Not necessarily. "Unable to find paid work" may just be an exaggeration on his part--he's done something to stay alive. Probably bit jobs here and there, with a bit of teaching, and maybe an apprenticeship to a teacher at some point. Tutoring. Scraping by. Never having a permanent position. Or losing that position as soon as they figured out he was a werewolf--that would also be a literal interpretation of the line. That as soon as he was "who he was"--a werewolf--he coulnd't hold a paying job.

However, in OotP, it seems like the real hardships came after his year at Hogwarts, as he has apparently seriously mouthed off to Sirius about Dolores Umbridge having written laws that "made it nearly impossible" for him to find work, and Harry reflects that he looks even shabbier. So he must have been doing something previously that he could no longer do.