When did Chonon and Admon live? Do we know any of their background?
In SEFER HA'YUCHSIN (cited by Seder ha'Doros) it is explained that Admon and Chanan lived at the end of the era of the second Beis ha'Mikdash, several hundred years after the death of Shimon ha'Tzadik. The Sefer ha'Yuchsin writes that they were particularly active in the year 3722 (28 BCE). However, it is still possible to explain, as we did in the Insights, that they were part of the Anshei Kneses ha'Gedolah, according to what Rashi writes in his commentary at the beginning of Avos, that the Anshei Kneses ha'Gedolah convened until the end of the days of the second Beis ha'Mikdash.
M. Kornfeld
See R' Reuven Margaliot's booklet the "Mishna VaArichata," in Berurim Bet about the difference between Anshe Keneset Hagedola and Sheyare Anshe Keneset Hagedola. Maybe Chanan was from the Sheyare Anshe Keneset Hagedola.