I believe there is an REO (bank owned) home in the Somerset Parkside condos for around $180,000 (near 11th & Q) but my company also has a ground floor unit listed in that complex for about the same price. The main drawback, as with all condos, is the association fee, which runs around $300 a month.
I also saw a home on 8th and D that was priced less than $250,000, but I don't remember the exact price. So there are definitely homes on the market right now that would fit your criteria. Be careful if you look at the condos at 200 P Street because many pick up traffic noise as some overlook the freeway. Some people don't mind looking at the freeway, but I can think of better views in downtown than that. :)