How can a person become a talmid chacham when he has to work to support his family? It would so difficult, since he is not given as many hours to learn as a kollel yungerman. But is it not possible somehow? What is the key? Proper review?
The Talmud in fact mentions that more people succeeded at becoming working scholars than at becoming just scholars:
ת"ר ואספת דגנך מה ת"ל לפי שנא' לא ימוש ספר התורה הזה מפיך יכול דברים ככתבן ת"ל ואספת דגנך הנהג בהן מנהג דרך ארץ דברי ר' ישמעאל ר"ש בן יוחי אומר אפשר אדם חורש בשעת חרישה וזורע בשעת זריעה וקוצר בשעת קצירה ודש בשעת דישה וזורה בשעת הרוח תורה מה תהא עליה אלא בזמן שישראל עושין רצונו של מקום מלאכתן נעשית ע"י אחרים שנא' ועמדו זרים ורעו צאנכם וגו' ובזמן שאין ישראל עושין רצונו של מקום מלאכתן נעשית ע"י עצמן שנא' ואספת דגנך ולא עוד אלא שמלאכת אחרים נעשית על ידן שנא' ועבדת את אויביך וגו' אמר אביי הרבה עשו כרבי ישמעאל ועלתה בידן כר' שמעון בן יוחי ולא עלתה בידן
Our Rabbis taught: And thou shalt gather in thy corn. What is to be learnt from these words? Since it says, This book of the law shall not depart out of thy mouth, I might think that this injunction is to be taken literally. Therefore it says, 'And thou shalt gather in thy corn', which implies that you are to combine the study of them with a worldly occupation. This is the view of R. Ishmael. R. Simeon b. Yohai says: Is that possible? If a man ploughs in the ploughing season, and sows in the sowing season, and reaps in the reaping season, and threshes in the threshing season, and winnows in the season of wind, what is to become of the Torah? No; but when Israel perform the will of the Omnipresent, their work is performed by others, as it says. And strangers shall stand and feed your flocks. etc., and when Israel do not perform the will of the Omnipresent their work is carried out by themselves, as it says, And thou shalt gather in thy corn. Nor is this all, but the work of others also is done by them, as it says. And thou shalt serve thine enemy etc. Said Abaye: Many have followed the advice of Ishmael, and it has worked well; others have followed R. Simeon b. Yohai and it has not been successful.
However, this may not be analogous to our society. Rambam mentions the schedule of what was apparently a regular working person in his time:
כיצד היה בעל אומנות והיה עוסק במלאכתו שלש שעות ביום ובתורה תשע
How so? If he was a craftsman and would engage in his work [for] three hours in the day, and [engage] in Torah for nine [hours]
It sounds like, a typical working scholar would actually spend the vast majority of his day learning rather than working. It is no surprise that a worker could be a scholar under such a regimen; that is probably the same amount of time a typical student spends learning per day in a full-time yeshiva program.
The very next line in the above Talmudic passage also indicates something along these lines:
א"ל רבא לרבנן במטותא מינייכו ביומי ניסן וביומי תשרי לא תתחזו קמאי כי היכי דלא תטרדו במזונייכו כולא שתא
Raba said to the Rabbis: I would ask you not to appear before me during Nisan and Tishri so that you may not be anxious about your food supply during the rest of the year.
This makes it sound like the working scholars were still spending the vast majority of their time learning.
However, with most modern professions it is not very practical to only work three hours per day, or only work two months out of the year. As such, we would expect it to be a lot harder to become a working scholar in these circumstances. Nevertheless, if you spend every free moment learning, you should be able to amass a considerable amount of time dedicated to learning.
It is interesting to note that The Talmud in fact asks a similar question:
תנו רבנן חסידים הראשונים היו שוהין שעה אחת ומתפללין שעה אחת וחוזרין ושוהין שעה אחת וכי מאחר ששוהין תשע שעות ביום בתפלה תורתן היאך משתמרת ומלאכתן היאך נעשית אלא מתוך שחסידים הם תורתם משתמרת ומלאכתן מתברכת
Our Rabbis taught: The pious men of old used to wait for an hour and pray for an hour and then wait again for an hour. But seeing that they spend nine hours a day over prayer, how is their knowledge of Torah preserved and how is their work done? [The answer is] that because they are pious, their Torah is preserved and their work is blessed.
These pious men of old spent as much time on prayer as a typical person might spend nowadays at work. The Talmud's answer to how they maintained their Torah knowledge with such limited time to devote to it seems to be that because of their piety they received special Divine assistance. Perhaps, then, a working scholar nowadays should strive to be especially pious.
Returning back to the first Talmudic passage, we find another related point:
אמר רבה בר בר חנה א"ר יוחנן משום רבי יהודה בר' אלעאי בא וראה שלא כדורות הראשונים דורות האחרונים דורות הראשונים עשו תורתן קבע ומלאכתן עראי זו וזו נתקיימה בידן דורות האחרונים שעשו מלאכתן קבע ותורתן עראי זו וזו לא נתקיימה בידן
Rabbah b. Bar Hanah said in the name of R. Johanan, reporting R. Judah b. Ila'i: See what a difference there is between the earlier and the later generations. The earlier generations made the study of the Torah their main concern and their ordinary work subsidiary to it, and both prospered in their hands. The later generations made their ordinary work their main concern and their study of the Torah subsidiary, and neither prospered in their hands.
The point here seems to be that the key to succeeding as a working scholar is to make sure that your Torah study is your main priority, and your work is just a side thing that you have to do.
גְּדוֹלֵי חַכְמֵי יִשְׂרָאֵל הָיוּ מֵהֶן חוֹטְבֵי עֵצִים וּמֵהֶן שׁוֹאֲבֵי מַיִם וּמֵהֶן סוּמִים וְאַף עַל פִּי כֵן הָיוּ עוֹסְקִין בְּתַלְמוּד תּוֹרָה בַּיּוֹם וּבַלַּיְלָה וְהֵם מִכְּלַל מַעְתִּיקֵי הַשְּׁמוּעָה אִישׁ מִפִּי אִישׁ מִפִּי משֶׁה רַבֵּנוּ:
Some of the great scholars in Israel were hewers of wood, some of them drawers of water, and some of them blind: nevertheless they engaged themselves in the study of the Torah by day and by night. Moreover, they are included among those who translated the tradition as it was transmitted from mouth of man to mouth of man, even from the mouth of Moses our Master.
He maintains similar sentiments in Laws of Gifts to the Poor 10:19.
The rabbis of the Talmud had jobs to support themselves. Hillel was a woodchopper and welldigger. Shammai was a building contractor. Rabbi Yochanan b. Zakkai was a businessman for forty years. Rabbi Yehoshua b. Chananiah was a blacksmith. Rabbi Meir and Rabbi Chananel were scribes. Rabbi Chanina sold bees' honey. Rabbi Chiyya the Elder, Rabbi Shimon ben Rabbi, and Rabbi Shimon ben Gamliel traded silk at Tyre [Gen. R. 77:2]. Rabbi Yochanan HaSandlar was a shoemaker (HaSandlar = The Sandal-Maker). Rabbi Yehuda the Baker was... guess what. Rabbi Yosi b. Chalafta was a tanner. Rabbi Yehuda ben Isaiah was known as 'the perfume-maker'. Rabbi Yehoshua was a Grit-Dealer. Rabbi Shimon P'kuli was a cotton dealer. Rabbi Shmuel b. Shilas was a school teacher.
Rabbi Safra and Rabbi Dimi of Nehardea were merchants. Rabbi Abba b. Zavina was a tailor. Rabbi Yosef b. Chiya and Rabbi Yannai owned vineyards. Rabbi Huna was a farmer and raised cattle. Rabbi Chisda and Rabbi Papa were beer brewers. Abba Chilkiyah was a field laborer. Abba Shaul was a gravedigger. Abba Hoshiya was a launderer and stomped olives with Rabbi Chiyya the Elder. Karna was a wine smeller (determined how long wine could last before going bad). Rabbi Chiya b. Yosef was in the salt business. Rabbi Eleazar ben Rabbi Zadoq and Rabbi Shaul ben Botnit were storekeepers in Jerusalem. Rabbi Eleazar ben Azariah dealt in oil and wine. Abba Bar Abba was a silk merchant. His son (Mar) Shmuel was a physician. Many were in the textile industry.
Same with the rishonim. Rashi was a vintner (grew vineyards and made wine). The Rambam and the Ramban were physicians. My rabbi growing up, Chief Rabbi of Egypt Hayyim Douek, was a businessman with gentile partners. That was the norm in the Sephardic world.
The Rambam did not mince words when describing those who do nothing but study and rely on charity for support:
Anyone... who makes up his mind to study Torah and not work, but live on charity, profanes the name of God, disgraces the Torah, obscures the light of religion, causes harm to himself, and deprives himself of life in the future world; for it is forbidden to derive temporal advantage from the words of the Torah. The sages said: “Whoever makes selfish use of the teachings of the Torah takes his own life.” They further told us: “Do not make of them a crown with which to magnify yourself, nor a spade with which to dig.” They urged us strongly moreover: “Love work, and hate arrogance.” “All Torah study which is not combined with some work must at length fail and occasion sin” (Pirkei Avot 2:2). The end of such a person will be that he will rob his fellow man. Anyone who maintains himself by the work of his hands possesses a great virtue, a virtue of the early saints, whereby he will attain all the glory and happiness of both this and the future world, as it is written [by the psalmist]: “When you eat the fruit of the labor of your hands, you shall be happy and it shall be well with you” (Psalms 128:2). “You shall be happy” in this world, “and it shall be well with you” in the future world, which will be altogether good. [Mishneh Torah, Sefer Ha-mada (Book of Knowledge) 10-11]
Others here have given sources which assure us that it is possible to become a talmid chacham while holding a day job, but without explaining how to do it. As I wrote in a comment, the question is really "How do I become a talmid chacham?" and then just squeeze that in whenever you can. Now, we often think that becoming a talmid chacham is accumulating book knowledge, and then people treat Torah study as a life-long cramming session for the Great Test Up Above. If this was true there would be no way to compete with anyone who learns all day.
The reality is that someone who stuffs themselves with book knowledge becomes a beast of burden with a load of books, not a talmid chacham. The correct goal is to develop a systematic internalization of the Torah, which requires a very different approach then learning as much as you can.
You do not become a talmid chacham by learning the whole Torah. You become a talmid chacham by developing your understanding until it reaches maturity. You should start with specific topics, and do them well, giving them the time they need. Think of it as planting a tree. You start with the seed, and it takes a really long time until it sprouts and starts growing, but after some years it is big and broad and leafy. You want your learning to develop like that.
You should choose the specific topics you start with partly by what interests you, and partly by what comes up in real life. This is the material you will pick up most naturally. Don't get stuck for too long with a narrow focus. Do the topic well, until you feel that you ahve a satisfactory understanding, and then move on to the next topic. Then come back, and do it again. And again. Each time, and some depth and breadth.
Wide knowledge is the 'fertile ground' your 'trees' grow in. You also need to take time to cover material quickly and superficially, just to be familiar with the terms and basic concepts, but do not mistake that for proper learning. You need to mix this broad knowledge acquisition (which also probably won't go that quick, especially as you start) with the focused learning which slowly develops from the core 'seeds'.
At some point -- this will take years -- your knowledge starts to integrate. The ideas you developed in one are connect to new material in other areas. Your knowledge starts to increase incrementally. Your 'trees' become a 'forest' of knowledge and understanding. This is when your learning start to mature. It will still continue to develop from here.
How do you do this while working? Its really up to you, but all it comes down to is just doing it. Figure out some time which is good for quicker and more suprficial learning. Maybe you can listen to shiurim on your commute. Also find when you can do deeper focused learning, maybe early in the morning or late at night. Do it even if it is only occasionally, but do it when you can.
Try to find a chavrusa if possible, even if it is an ocassional meeting to review some in-depth material. Try to find a mentor.
Learning is very personal. You have your own specific path to becoming a talmid chacham, which you need to discover. Take suggestions from others, look at what others do, but then think about what will be good for you. This is about you, not about following recipes written by others. Find your own path to becoming the talmid chacham that you have the unique personal potential to be.