Did Shem's physical body (or) Shem's teachings remain alive 600 years?

Question

Did Shem's physical body (or) Shem's teachings remain alive 600 years?

• What is the correct understanding of וַֽיְחִי־שֵׁ֗ם אַֽחֲרֵי֙ הֽוֹלִיד֣וֹ אֶת־אַרְפַּכְשָׁ֔ד חֲמֵ֥שׁ מֵא֖וֹת שָׁנָ֑ה וַיּ֥וֹלֶד בָּנִ֖ים וּבָנֽוֹת from Bereishit 11:11?

• If Shem's physical body remained alive 600 years, then Shem may have literally seen and spoken with Avram, Yitschaq at age 538. This would mean Shem possibly met Yaaqov before passing away.

• Does the 600-year lifespan actually mean Shem's teachings remained with his descendants?

Does Judaism believe Shem's life was only physical (120-years) & influential (480-years)? -- [based on Bereishit 6:3]?

Answer 1

There are a number of sources that point to the fact that Shem did indeed live a long life and also lived to see some of the personalities that you mention above.

Firstly by way of introduction, the Gemara in Bava Basra 121b highlights Shem as one of seven people, whose longevity in physical years collectively spanned the history of mankind. It writes there:

תָּנוּ רַבָּנַן שִׁבְעָה קִפְּלוּ אֶת כָּל הָעוֹלָם כּוּלּוֹ מְתוּשֶׁלַח רָאָה אָדָם שֵׁם רָאָה מְתוּשֶׁלַח יַעֲקֹב רָאָה אֶת שֵׁם עַמְרָם רָאָה אֶת יַעֲקֹב אֲחִיָּה הַשִּׁילוֹנִי רָאָה אֶת עַמְרָם אֵלִיָּהוּ רָאָה אֶת אֲחִיָּה הַשִּׁילוֹנִי וַעֲדַיִין קַיָּים

§ After discussing the generation of those who died in the wilderness, the Gemara mentions a tradition that relates to that generation. The Sages taught: Seven people spanned in their lifetimes the whole world in its entirety, i.e., their lives have spanned all of human history. Methuselah saw Adam in his lifetime; Shem saw Methuselah; Jacob saw Shem; Amram saw Jacob; Ahijah the Shilonite saw Amram; Elijah saw Ahijah the Shilonite; and Elijah is still alive. (Sefaria translation & notation)

Indeed with regards to this point, the Seder Olam Rabbah 1 notes:

ושם שמש את מתושלח צ"ח שנה

And Shem served Mesushelach for 98 years.

In Tanna D'vei Eliyahu Rabbah 24, it highlights a solid span of time that Shem was engaged in nevuah (prophecy), trying to curb the ways of the ovdei avoda zara (idol worshippers):

ולא זו בלבד אלא שם הגדול בן נח שנתנבא ארבע מאות שנה על כל אומות העולם ולא קבלו ממנו

And not only this, but the great Shem, son of Noach, prophesied for 400 years to all the nations of the world and they did not accept him.

As far as meeting Avraham, it is clear that they enjoyed several interactions and @Harel13 has already alluded to some of these in the comments above. Another example is that Avraham sent for Shem when he required bris milah (circumcision) and Shem performed the bris on both Avraham and his son Yishmael (refer to Yalkut Shimoni, Bereishis 80)

Shem definitely outlived Avraham, and the Midrash in Bereishis Rabbah 62 mentions the fact that he helped bury him. The Midrash there writes:

אָמַר רַבִּי שְׁמוּאֵל בַּר נַחְמָן שֵׁם וְעֵבֶר הָיוּ מְהַלְּכִין לִפְנֵי מִטָּתוֹ וְרָאוּ גַּבָּהּ מָקוֹם מֻפְנֶה לְאָבִינוּ אַבְרָהָם וְקָבְרוּ אוֹתוֹ בִּדְיוֹטְרִין שֶׁלּוֹ בַּמָּקוֹם הַמּוּכָן וּמְזֻמָּן לוֹ

Rabbi Shmuel son of Nachman said: Shem and Eiver were walking before his (Avraham's) bier, and they saw that the space (next to Sarah's grave) was free for Avraham, and they buried him in the spot that had been destined (lit. prepared) for him.

The Midrash HaGadol, Bereshis 11:11 definitively writes that he lived through a substantial part of Yitzchak's life too and the first third of Yaakov's life:

ויחי שם - חיה שם עד שעבר מימי יצחק מאה ועשר שנים והיה יעקב באותו הפרק בן חמשים שנה ובאותו זמן מת שם בן נח

And Shem lived - Shem lived until 110 year's of Yitzchak's life had passed, and Yaakov during this period was 50 years old, and at that time Shem the son of Noach died.

So to return to the question, yes - Shem lived in physical years for this amount of time, during which time he outlived Avraham, and lived whilst both Yitzchak and Yaakov were alive. The Seder Hadoros HaKotzair p.7 places the year of birth at 1558 and his death at 2158.

Answer 2

Seder Olam sets up the timeline of the world based on the life spans as shown in the torah. The Torah says that Shem was 100 years old two years after the Flood and lived 500 years after that. Thus Shem actually lived 600 years (500 years after the birth of his first son). The normal lifespan was then gradually reduced after the flood. Shem was 100 years old in 1658 and died in 2158. Avraham was born in 1948, Yitzchak in 2048, and Yaakov in 2108. Thus, Shem died when Yaakov was 50 years old, before Yaakov left his parents for the Yeshivah of Ever.

Rashi on Bereishis 6:3 explains that the 120 years means the time until the flood was going to start.

and his days shall be: Until a hundred and twenty years I will delay My wrath towards them, but if they do not repent, I will bring a flood upon them. Now if you ask: from the time that Japheth was born until the Flood are only a hundred years, [I will answer that] there is no [sequence of] earlier and later events in the Torah. This decree had already been issued twenty years before Noah begot children, and so we find in Seder Olam (ch. 28). There are many Aggadic midrashim on the words לֹא יָדוֹן, but this is its clear, simple explanation.

The normal lifespan was reduced gradually after the flood.