What could women in the Jewish world do in public? This is often overstated in order to dramatize the difference with Jesus and the early churches. So a good look at some of the evidence may help all of us.
First: Commandments and Religious Service: women are excluded from commandments pertaining to religious service “because women are not obligated to fulfill commandments” (tBerakot 7:18; bMenahot 43b). But, though attempts were made, no clear operative rule developed to discern which commandments were expected (or not expected) from women. However, the reality is that women performed many religious commandments: prayer at the Temple (Lamentations Rabbah 3:3), sacrifice after the birth of a child (mKeritot 1:7), pilgrimages to the Temple (Luke 2:41-50; bNedarim 36a; mMiddot 2:5), nazirite vows (mNedarim 6:11; mNazir 3:6), sometimes perform circumcision on her son (1 Maccabees 1:60-61; 2 Maccabees 6:10; 4 Maccabees 4:25; tShabbat 15:8), wearing phylacteries (Mekhilta de R. Ishmael Bo 17), establishing an abode by depositing foodstuffs (erub; yErubin 3:2, 20d).

Second: Occupations and Professions: the fundamental responsibility of the mother was rearing the children. The father’s responsibility was to circumcise (but see 8.1.2), teach the boy Torah, redeem him, and marry him off (bQiddushin 29a) and the mother was exempt from these things (mQiddushin 1:7). Age seems to be a critical factor here. The mother’s tasks revolved around household chores: baking, cooking, cleaning, sewing, washing. The husband’s tasks were outside the home (mKetubot 5:5).
Reality was different: women could turn household duties into a business (selling dough, garments; cf. mHallah 2:7; mBaba Qamma 10:9), could be used to make things for the Temple (tSheqalim 2:6), and they could be shopkeepers (tBaba Qamma 11:7; bBaba Qamma 119a; mKetubot 9:4; bKetubot 86b; mShabuot 7:8) and innkeepers (tDemai 3:5; 4:32; mYebamot 16:7). Women were also hairdressers (mKelim 15:3; mQiddushin 2:3; later rabbis interpreted “Mary Magdalene” as a hairdresser [Heb., gdlt] at bShabbat 104b; bHagigah 4b), midwives (mShabbat 18:3), various medical professions (yShabbat 18:3, 16c; tShabbat 15:15; see esp. Wainright), and professional mourners (mMoed Qatan 3:9). It was, therefore, common for women to work.
What of Women as Merchants? It is possible that some Jewish women were merchants and traveled for business. Sawicki argues that Joanna and Mary the Magdalene were business associates and “hired” Jesus for Roman guests to Tiberias.
Third: Study of Torah: Josephus brags that Jewish women knew the Torah (Against Apion 2:181). Some evidence suggests women and daughters were not taught the Torah but that evidence seems to be concerned more with providing ammunition against sexual promiscuity (mSotah 3:4; ySotah 3:4; 19a; bYoma 66b). Writing and reading were not common among Jewish peasant women; the royal women did read and write (Josephus, Antiquities 15:24, 45, 62; 17:134-141; War 1:641). But women were expected to be able to write a bill of divorce (mGittin 2:5).
Some evidence suggests women did study the Torah (mNedarim 4:2-3; tBerakot 2:12; tKetubot 4:6-7). One woman was criticized for neglecting her household duties to study the Torah (ySotah 1:4, 16d; cf. Luke 10:38-42). Sometimes women’s knowledge was needed to determine halakhah (cf. bNiddah 13b; 48b; yShabbat 4:1, 6d). Two women became known, legendarily, for Torah knowledge: Beruriah (bPesahim 62b) and Matrona (Genesis Rabbah).
Additional points:
Women could be members of “sectarian movements” (cf. esp. T. Ilan, “Attraction”).
Women could be prophets (cf. 2 Kings 22:14-20; Nehemiah 6:14; Acts 21:9; 1 Corinthians 11:5; and possibly John 11:27).
Footnote: Recently a Spanish scholar has argued that “possession by demons” is a form of social protest; female possession at the time of Jesus was a social protest within the norms of male-structured society against patriarchy (see Ubieta).