Anshei Knesset HaGedolah were the 120 sages set up by Ezra at the end of the Babylonian exile. According to shiurim that I have attendednce it was set up, no more members were added as the original members died until only Shimon Hatzadik was left. This is from a number of sources (including le'havdil Wikipedia) as well as a number of Jewish History sites and text books.
Rabbi Berel Wein discusses them in "Echoes of Glory" pages 13-15. Rabbi Reuven Margolies in Yesod Hamishnah V'Arichasa (Mosad Harav Kook, 1956) suggests that there were never any replacements and Shimon was the sole survivor of the 120. Others interpret "from the remnants" in Pirkei Avos as meaning that he was the one who picked up the mesorah after the group disbanded.
Pirkei Avos 1:2 shows that in the chain of tradition that Shimon Hatzadik was the "from the remnants" this group and was the next link in the chain.
Note that there are disputes as to the precise date that the first temple was destroyed and the second temple rebuilt. This is a totally different matter than the actual existence of the Anshei Knesses Hagedolah as there are those who say that the first temple was destroyed in 586 BCE and others as late as 420 BCE. This is a question discussed by Rabbi Shimon Schwab and others. One reason for this difference is the fact that the second temple traditionally existed for 420 years with 70 years of exile. Counting back from 70 CE gives 420 BCE. The discussion depends on how the calendar was set up in the construction interval and when the 420 years is counted from.
See @Matt in his answer as well for other citations for more details especially the Where and When section.
Obviously, those who say that it lasted from 520 BCE to 200BCE do not believe that it was a one generation organization and place Shimon Hatzaddik later would not agree with Rabbi Margolies.
The following is a short description of this group.
History Crash Course #26: The Great Assembly
The Men of the Great Assembly -- in Hebrew, Anshei Knesset HaGedolah --
was an unusual group of Jewish personalities who assumed the reigns of
Jewish leadership between 410 BCE and 310 BCE. This time period
follows the destruction of the First Temple, and includes the early
decades of the Second Temple, up until the invasion of the Greeks, led
by Alexander the Great. Realizing that the Jewish people were growing
weaker spiritually, a group of wise leaders came together -- expanding
the Sanhedrin, the Jewish Supreme Court, from 70 to 120 members --
with a special aim of strengthening Judaism. Initially gathered
together by Ezra, they defined Judaism in this tumultuous time when
prophecy and kingship were all but gone from the Jewish people.
(Today's Israeli Parliament, which is called "the Knesset," also has
120 members in imitation of the Great Assembly although the Knesset of
today serves an entirely different function of the Great Assembly of
2,500 years ago.)
Among them we count the last of the prophets Haggai,
Zechariah and Malachi, as well as the sages Mordechai, (of the Purim
story), Yehoshua, (the High Priest), Nechemia (the chief architect of
rebuilding of Jerusalem), Shimon HaTzaddik (also a High Priest).
Anshei Knesset HaGedolah
“Anshei Knesset HaGedolah” – Men of the Great Assembly; founded by
Ezra in approximately 520 B.C.E., this institution of Torah Sages led
the Jewish People at the beginning of the Second Temple Era (ca. 520
B.C.E. – 70 C.E.). It included Mordechai and the last of the
prophetsChaggai, Zechariah and Malachi.
Among the accomplishments of the “Anshei Knesset HaGedolah” were
finalizing the contents of the “Tanach,” the 24-Book Hebrew Bible,
instituting the “Shemoneh Esray” Prayer (recited at least three times
daily, and ultimately to serve as a substitute for the Temple
Sacrifices), and the enacting of many Laws to protect and bolster the
observance of the Torah Commands.
According to Pirkei Avot (1:1), they are the fifth link in the Chain
of Jewish Tradition: 1) Moshe receives it from Sinai and teaches it to
2) Yehoshuato the 3) Elders to the 4) Prophets to the 5) Anshei
Knesset HaGedolah, at the end of the Biblical Period (ca. 520 B.C.E.).
Pirkei Avot (1:2) also identifies “Shimon HaTzaddik,” Shimon the
Righteous, as “among the last of the Men of the Great Assembly,” at
the beginning of the Talmudic Period (ca. 200 B.C.E.).
In truth, the “Anshei Knesset HaGedolah” was a transitional
institution, that over the approximately 320 years of its existence
guided the Jewish People from the Biblical Period to the Talmudic
Period, from the Period of “Nisim Niglim,” open, revealed miracles,
observed by the entire People, as were the miracles associated with
the Exodus from Egypt and the splitting of the Sea of Reeds, and the
Miracle performed by Eliyahu HaNavi in his contest with the Prophets
of the Baal, to the Period of “Nisim Nistarim,” to more modest and
hidden, concealed miracles; as were the Chanukah Miracle of the Oil
and the “hidden” miracle of Purim. From the period of “Nevuah,”
Prophecy, to the Period of “Tefilah,” Prayer. In Chassidic terms, it
was a transition from a Period of “Isarusa Mil’ela,” Arousal from
Above, to a Period of “Isarusa Mil’tatoh,” Arousal from Below.
There was a sense among Chaza”l that the “Beit HaMikdash HaSheni,” the
Second Holy Temple, would not last, because the Divine Presence was
not as concentrated as it had been in the First Temple, and it would
be necessary to prepare the People for a long, uncharted journey in
theDiaspora, with only the guiding but unseen “Hand of HaShem,” and
His “Eyes,” watching from behind the curtain.
Thus, this institution was called “Great” because it “restored the
Crown of the Torah” (Yoma 69b and Berachot 33a), served as the
spiritual center of Jewish Life for approximately 320 years, and
ensured the survival of the Jewish People through the coming harsh
conditions of the Diaspora until the arrival of Mashiach ben David,
soon and in our days.