I'm from Upstate NY, AMA. Jamestown is a bleak place, even by Upstate NY Standards (though Jamestown is considered a part of the Southern Tier, which is even worse off). That said, $4,000 is a ridiculous price for a house. On the other hand, I can't see much of a market for any house there, and that's really what dictates the price of it.
I wonder what the property taxes are. Here in Detroit you can get houses for $500, but the property taxes are actually quite high, so there's not much sense in scooping a bunch up. For a long time, I've thought that the US could offer Urban Renewal visas, and basically homestead these properties to immigrants. If they live in the neighborhood for 8 years or so, they can get a greencard and own the property. In a city like Detroit where the population and tax base has been in decline, it might be useful.
I don't know much about property taxes so maybe you'll be able to gleam more information from the Jamestown assessment page than I can. I wouldn't imagine the taxes there are too high. Southern tier of NY is sparsely population and not near much unless you're into camping and that sort. The lone exceptions being Ithaca and Binghamton.
Interestingly, one of the reasons why Detroit's property taxes are high is because it is sparesly populated. As many people with wealth left the city, the tax base collapsed, and one of the options was to raise the rates on the remaining residents. Because crime is high and many lack insurance, auto insurance rates in Detroit are extremely high too. It's definitely a case of paying more for less.
I'm also originally from upstate New York (Utica). I can tell you about another lovely side effect of raising the taxes on the population that doesn't have the means to flee: arson! Thanks to an ancient tax assessment structure, there were a lot of apartment buildings being taxed each year close to the cost of the entire structure. Oh sure, it seems great to buy a three-bedroom house for $30k but the annual tax could be $6k. A lot of owners could not sell these places, so they started torching them. How bad did it get? 1) I knew a couple kids that ran a simple business. For $50 they'd set up your undumpable dump so that you could torch it easily without losing the insurance claim. For example they'd shave the wire on a toaster a certain way so it'd fray and catch fire. They'd explain, "you'll have 20 minutes to get out after to plug it in and put in the toast." 2) They called in the National Guard in 1998 to fight arson. 3) The mayor responded by trying to close fire stations. He was so oblivious to the real problems that he hoped to make the fire department a volunteer organization. This was illegal for any municipality larger than 50k residents in the state, so that failed. 4) Some of the firefighters got in on the arson to guarantee their jobs. So yes, a house that costs less than a new car loan sounds wonderful... until you realize you're the only person paying for street light and paving in the entire neighborhood.