Zanesville has a cool
three-way bridge that Amelia Earhardt said made it the "most recognizable town in the country", and is known for its pottery industry. Although that industry was larger in the past. It's also close to the Hocking Hills area of southeast Ohio, known for its natural beauty. Wouldn't be in my list of top 5 places to move in Ohio, but if you could find a decently safe neighborhood, and with the scenario set up of $20K/year and free rent, you could probably do worse. In this scenario I'd probably buy a cheap used Corolla and focus on the bountiful outdoors opportunities.
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The worst area to choose in my neck of the woods is generally regarded as East Cleveland, Ohio, not to be confused with the eastern part of Cleveland, Ohio. Three of its last four mayors have been convicted of crimes, the current one has been indicted for four felonies, and it has lost half its population in the last 20 years, an impressive rate even by Rust Belt standards. It has a 41.8% poverty rate and is the
4th-poorest city in the United States, with an average household (not individual, but household) income below the UBI threshold that Zardnaar chose. The city itself is known for being so broke that many local lawyers refuse to file suit against it, even with a good case (which sometimes exists given the corruption), due to the unlikelihood of the city having enough money to pay out to settle. On the upside, it has good public transportation connections to Cleveland, so you could easily travel to Cleveland itself and access the big city's amenities.
You could make a case for East Cleveland having some potential for the future given its transit infrastructure - the commute to downtown is a fairly short and relaxing train ride, so in theory it could develop as a commuter suburb in the future - but given how many of its residents have left in so little time, I can't argue that the upsides outweigh the downsides.
As for the hypothetical, I think one key question is to define the parameters for "decent house in a bad area with free rent". A more interesting question might be, "if you had a $25K basic income, where would you live?", with a goal of finding the best, rather than least desirable, candidate.