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It's pretty much a sliding scale, so you can by Irish Tool
pick a target price or mileage that works for you and then find something accordingly. Personally I would be leery of any car under $5,000, and mileage is pretty wide open but I would be comfortable buying anything up to 100k miles if it was known as a reliable model and has had regular maintenance.
So another poster said he gets 70k+ mile cars for around $6k; I got a used economical car at 25k a few years ago for around $11k... I'd say it's pretty much a sliding scale between those numbers with some variation for how much the original car cost. Personally I wanted low mileage because I knew I would be putting on a ton myself, so that was my primary target.
One thing to be mindful of on higher mileage cars is when the timing belt is supposed to be changed. It's probably the most typical costly maintenance item that has to get done on schedule. Tires would be the other one (other parts that can go bad eventually, like the transmission, aren't as predictable/regular). People will often sell a car about a year before they'd have to change the timing belt or tires.
You can look up at what mileage any car is supposed to have its belt changed via Google, and then keep that in mind while shopping. Also any seller should be able to tell you how many miles are left on the tires (there are ways to tell visually if you know what you're looking at, but usually the tire maker provides a good estimate of the tire's life when sold, so if the owner has a receipt from the tire purchase you can also have a good estimate from that).
Also, buy Japanese for reliability. Or maybe some of the recent Fords.