2000 years ago in the Roman occupied Jewish world, people who were called rabbi generally had real jobs on the side.
There were no organized rabinnic seminaries back then, but leading rabbis generally had disciples (Hillel and Shammai had many), and a rabbi could ordain his students when he thought they had learned enough. Whether that ordination was accepted by the larger community depended on the reputation of the ordaining rabbi. (Ordination still works that way in some corners of the Orthodox Jewish world, and also in some corners of the evangelical Protestant world. The rest of us, Jews and Christians, rely on organized seminaries for our clergy.)
What follows is a list of some of the first and second century rabbis from the Talmud and their "day jobs." It's extraordinarily likely that Jesus knew some of these men (the statement of the Golden Rule in the Gospels is very similar to Rabbi Hillel's formulation), and if Jesus was a carpenter and rabbi that mix is merely typical of the other rabbis we know from the period.
- Hillel -- woodchopper
- Shammai the elder -- builder or carpenter or something
- Abba Chilkiyah -- field laborer
- Yochanan ben Zakkai -- some kind of businessman
- Abba Shaul -- gravedigger (considered a dishonorable job)
- Abba Oshiya -- laundry worker of some kind
- Shimon Pkuli -- cotton? flax? dealer
- Shmuel ben Shilas -- school teacher
- Mier -- scribe
- Channai -- scribe
- Yosi ben Chalafta -- tanner (considered a dishonorable job)
- Yochanan Hasandlar -- as the name suggests, shoemaker
- Yehoshua ben Chanania -- blacksmith
- Safra -- merchant
- Dimi of Nehardia -- merchant
- Abba ben Zavina -- taylor
- Yosef ben Chiya -- vintner, either vinyard owner or wine maker
- Yannai -- vintner
- Huna -- cattle farmer
- Chisda -- beer brewer
- Papa -- brewer
- Karna -- wine merchant
- Chiya ben Yosef -- salt maker or dealer
- Abba bar Abba -- silk merchant
- Mar Shmuel -- physician
K-HB asked for sources. Most of the information we have about the Jewish commuity of the era is found in the Talmud, and like the New Testament, this is all material that was put down on paper years after the events, so we're relying on oral transmission for a while before it was written down. That, and the sheer volume of text in the Talmud are both barriers. What we know about the rabbis mentioned in the Talmud comes mostly from stories about them that are included in the text as illustrations of one point or another.
For example, in Tractate Shabbat, page 31a (citations are traditionally abbreviated, so this would be cited as Shabbat 31a), there's a story about a Gentile convert who mockingly approaches Rabbi Shammai saying "I'll convert to Judaism if you can explain the entire Torah while I stand on one foot." Rabbi Shammai responded by beating him with his ruler (literally, builder's cubit). The story goes on to say that the same guy went to Hillel with the same mocking question. Hillel responded "that which is hateful to you, do not do to your fellow, that is the whole Torah, the rest is commentary. Now go and learn." The point of the story was to illustrate the difference between the two Rabbis, but we can glean from it that Shammai had a ruler of the type you'd expect a carpenter or builder to carry. Similar logic covers all the others, and there are lots of lists you can Google that give such lists of "day jobs"