There is no clear source about the maximum length of hair known to me, but it's possible to glean information from some related topics.
The Mishnah (redacted c. 200 in Roman Palestine), based on Leviticus 19:27, requires a minimum permissible length for the sideburns of the hair, but not a maximum (Makkot 3:5, BT 20b). The fact that a minimum was defined and not a maximum might be due to the fact that closer haircuts were more fashionable, but it could just be exegesis of the Biblical prohibition.
The bangs (and a yearly ceremonious cutting of them) are mentioned as an idolatrous practice in the Mishnah (Avoda Zara 1:3), or an "Amorite" custom in the Talmud (BT Sota 46b, attributed to a Galilean sage, not a Babylonian). This seems to reflect a fashion for bangs among non-Jewish residents of Roman Palestine, to which Jewish residents were contrasted.
Curling hair (which might indicate a long length of hair) was described as childish (Tanhuma, Vayyeshev 7 about Joseph), and the servant of Rabbi Judah the Prince said to someone, "Until when will you curl your hair?" (BT Rosh Hashshana 26b). Perhaps children had longer hair than adults.
Some of these sources are quoted in this article about hair in Jewish sources by Miriam Samuel (Hebrew).
While these sources mostly point to short hair for adults, it should also be noted that while women's dress was forbidden to Jewish men (based on Deuteronomy 22:5), there is no discussion of long hair in this context in rabbinic sources, which might also show that it wasn't entirely unusual for men to have long hair. However, an explicit statement on the subject can be found in the writings of Paul the Apostle (I Corinthians 11:14-15), who views short hair as natural for a man and long hair as natural for a woman (however, while Paul was in Jerusalem for some time, he was from Tarsus and didn't stay in Judea).
One possible contemporary picture of a Judean is one bust identified, probably incorrectly, as that of Josephus, who was born in Judah and later moved to Rome. Even if Josephus is depicted, it's unclear whether the hairstyle reflects a Judean fashion, or if it was changed to match his later Roman neighbors, but the hairstyle does in fact have a close haircut with no bangs. (I left this paragraph here because it appeared it a previous version of the answer, but the source linked to above claims the identification with Josephus is baseless.)