As it happens, I'm now in the middle of reading a book called "Mevo Lekitvei Hakodesh" (Introduction to the Holy Scriptures) by Shmuel Shrira, which discusses the creation and canonization of the Scriptures, but there are many books on the subject1. There's also an excellent lecture by Rabbi Prof. Shnayer Leiman on the subject.
Naturally, as with just about every issue on the subject of dates and Tanach, there are disagreements both among our sages and among academic scholars. According to Rabbi Leiman, by academics there's a general consensus, though, that the Torah portion of the Tanach was already established and canonized by the time of the Jewish/Samaritan schism, because the Samaritan biblical canon consists of only the Pentateuch. This is surmised to have happened around the 2nd century BCE (Wikipedia puts the date as 122 BCE).
As for the rest of Tanach - the Nevi'im and Ketuvim sections - debates were still going on in the time of Rabbi Akiva, circa the 2nd century CE. We know this because of the Talmudic records on the matter:
For example, in Shabbat 30b it says:
"Rav Yehuda, son of Rav Shmuel bar Sheilat, said in the name of Rav: The Sages sought to suppress the book of Ecclesiastes and declare it apocryphal because its statements contradict each other and it is liable to confuse its readers. And why did they not suppress it? Because its beginning consists of matters of Torah and its end consists of matters of Torah. The ostensibly contradictory details are secondary to the essence of the book, which is Torah."
And in Mishnah Yadayim 3:5:
"All the Holy Scriptures defile the hands. The Song of Songs and Kohelet (Ecclesiastes) defile the hands. Rabbi Judah says: the Song of Songs defiles the hands, but there is a dispute about Kohelet. Rabbi Yose says: Kohelet does not defile the hands, but there is a dispute about the Song of Songs. Rabbi Shimon says: [the ruling about] Kohelet is one of the leniencies of Bet Shammai and one of the stringencies of Bet Hillel. Rabbi Shimon ben Azzai said: I have received a tradition from the seventy-two elders on the day when they appointed Rabbi Elazar ben Azariah head of the academy that the Song of Songs and Kohelet defile the hands. Rabbi Akiba said: Far be it! No man in Israel disputed that the Song of Songs [saying] that it does not defile the hands. For the whole world is not as worthy as the day on which the Song of Songs was given to Israel; for all the writings are holy but the Song of Songs is the holy of holies. If they had a dispute, they had a dispute only about Kohelet. Rabbi Yohanan ben Joshua the son of the father-in-law of Rabbi Akiva said in accordance with the words of Ben Azzai: so they disputed and so they reached a decision."
Some also surmise that there were debates on the canonicity of texts that today are known as the Apocrypha or "Sefarim Chitzonim in Hebrew. One reason that some say this is that, for example, Ben-Sira is quoted a few times in the Talmud by the sages, such as in Sanhedrin 100b and Ketubot 110b where it seems they used Ben-Sira as an "asmachta" much like when verses from Tanach are used.
Interestingly, as Rabbi Leiman states (can also be heard in this audio clip), Jerome, author of the Latin translation of the Bible, the Vulgate, lived in Beit Lechem at the time of the Amoraim and writes in his commentary on the Bible that he sometimes went to consult with the local Jews and specifically, the local rabbi - likely a certain Amora. Jerome wrote in his commentary on Yirmiyahu 29:22 that he was told by the local Jews that the two wicked men in this verse are the people Daniel interrogated according to the Book of Shoshanah (Susanna), except that Shoshanah could not have been a contemporary or accurate description of the events for one simple reason: Jews in exile didn't have the power of capital punishment (emphasis mine):
"The Hebrews say that these men who "committed foolishness in Israel" and "committed adultery with the wives of their fellow citizens" are the elders to whom Daniel spoke...But what is said in the present passage, "whom the king of Babylon roasted in the fire," appears to contradict the historia of Daniel2, which asserts that the elders were stoned to death by the people as a result of Daniel's judgement, whereas here it is written that the king of Babylon roasted them in the fire. For this reason, this story is rejected as a mere fable by many of us and by almost all of the Hebrews; nor do they read it in their synagogues. "For how could it be," it is argued, "that captives had the authority to stone their own leaders and prophets?" (Source)
Clearly, by the time of Jerome (mid 4th to early 5th century CE), Shoshanah had already been rejected by the majority of Jews (although apparently some still clung to it), but it seems that there was a time in which it may have been included within Daniel (see first footnote).
All of this is to say, that during the time of the DSS cult(s), the Torah was just about canonized (although there were still debates on certain specific phrases, for example, the Torah scrolls with variant phrases found in the Temple in Sofrim 6:4) but the rest of the Tanach - the Nevi'im and Ketuvim - were still being debated, and naturally, canonicity shifted between sects. And so, as rightly stated in the comments of your question, Appiah was referring to the Bible in its totality, not just to the Torah.
1 As @DrShmuel mentioned, Rabbi Reuven Margaliyot's Hamikra Vehamesora (can also be purchased here) is an excellent book that deals, among other things, the development of the Masoretic text.
2 If I remember correctly, in the LXX, Shoshanah is included in Daniel and not as a separate book.