Apparently whatever you want if you don't wind up in a penal colony. The Federation is a society where one can more easily self-actualize because of it's nature where hunger, poverty, disease, etc. have been wiped out. It's worth noting that civilian quality of life is on par with those in Starfleet; civilians also have access to modern technology like FTL travel, replicators, transporters and even holosuites, though they are presumably less powerful than their Starfleet counterparts.
Probably the same think you do if you're not a government employee today: whatever the hell you want. Since the Federation doesn't have a currency-based economy, it seems reasonable to say that occupation isn't a prerequisite for survival like it is today (even in wealthy nations like the US or UK), so a person could do wahtever they are interested in doing. Picard states that people do what they do for the betterment of others... which makes me wonder why there are so many people volunteering for dirty jobs like dilithium mining and the like.
^^ They don't have currency, but things like transporter credits and replicator rations have been mentioned in Trek. I'd also wager they have Holodeck credits as well to compensate people for work that is considered dangerous, hazardous, or specialized.
By the time of Voyager, it's established that holograms do the real dirty work.
- These are mentioned as ways to ration critical resources... on Starfleet ships. Since just about everything else seems to be free, why would people have to work for time on the holodeck?
Since the population of the Federation is perhaps as much as 1.5 trillion, the vast, vast majority of people are not in Starfleet, even if Starfleet is huge. People involved in the economic, governmental and cultural endeavours of Federation member worlds (the Federation may not have bills and coins but there are certainly media of economic exchange in use). Sure, a few incorrigibles in prison colonies. And, Federation member worlds probably have quasi-Starfleets. Commercial space travel has been depicted as early as ST:ENT "Horizon" ProfessorTrek 00:14, August 5, 2011 (UTC)
Join up! Become a citizen!
Open a Resteraunt- i.e. Joseph Sisko
Chair a University-i.e. Data
Join a Merchant Service- i.e. R.M.Merik
Tend a vinyard- i.e. Robert Picard
Supervise the Atlantis Project- i.e. Louis Blanchard
Write books- i.e. Jake Sisko
Be a research scientist- i.e. George Kirk
Actually, the Atlantis project might have been under the auspices of Starfleet, with Louis Blanchard just happening to be a civilian. But this is not canon ProfessorTrek 19:35, August 17, 2011 (UTC)
There is no reason the Atlantis Project would be controlled by Starfleet. The project was an internal project on Earth and would be controlled by either a science agency or the local government.