Oh boy, is this a blast from the past question. :D Fortunately, we do have answers for this. Part of the annoying bit is that some of the information is outside the story but Viz has not been good at translating and bringing this extra material to English. Let me try to make this short or it'll end up in my draft folder and I'll never get back to it.
As we know, three years ago, a jobless Saitama saved a boy from a monster, and in the process decided to forget about the job hunt and focus on being a hero. How did that work out for him? Well, from the 'Road to Hero' OAV, it saw him living in a terrible apartment block where the landlady tolerated his not paying the rent because she made her money off the criminal gang in there. After Saitama caught them to turn in for the police reward, she threw him out. Fortunately, the tailor who'd do his clothing repairs found him and hooked him p with the place we first met him living.
There's a reference to the shitty apartment block in the manga. Saitama is disgusted that her place got spared in the meteor shower. He may be a bit bitter, lol.
Anyway, you asked what did Saitama do for a living? Well, in the still-not-translated-into-English Hero Data Book, ONE explains that he did whatever part-time jobs and day work he could come by, lived very frugally, and was happy to take food handouts from grateful people.
If you do feed Saitama, he prefers veggies. Getting a year's supply of sweets was rather bad for his teeth, you see. That's from the bonus chapter in Volume 2, "Brushing up"
There have been a number of audio stories featuring the jobs Saitama has done, none officially translated to English, but a couple that there are fan translations of feature him working in construction (and accidentally helping Amai Mask workshop a new song while desperately needing a shit) and working in a carnival display (and accidentally giving Tatsumaki a fear of ghosts). ONE also did a sketch of Saitama trying to scan the knife a robber was trying to threaten him with:
All very well, but given that Saitama doesn't stick with any of these low-paying jobs for long, quitting as soon as he has enough money to live off of for at least a few weeks, how does he afford rent and utilities?
The part of town Saitama lived in had been abandoned due to the frequent monster sightings, but for some reason, the water and electricity supply to that area wasn't cut off, something Spring Mustachio remarks on in the chapter 'Rumors'
Genos doesn't appreciate Fubuki dropping the truth bomb on the issue of rent, but Saitama himself is proud to pay nothing.
According to ONE (again OFF-STORY!), Saitama hasn't spent the rent money Genos has given him. Considering that what he's really rented Genos is his right to privacy, he should.
In any case, since Saitama started working as a pro-hero, he has been getting paid. There's actually a whole bonus chapter on hero renumeration, 'What Can't Be Bought', the bonus chapter in Volume 5. Short version: they get paid like shit, and then only at the end of the month. The HA takes full advantage of the fact that heroes can't complain without looking mercenary.
Since those hard times before the first paycheck, Saitama has started getting paid, so he's got income. Also, his new place in the HA headquarters is officially free with the job, so life is almost cushy for Saitama.
Hope that answers your question!