It all depends on the dealer. I bought a used car once and it came with a 6 month warranty. At that time the car was 8 years old, so it was far from new. My last car was 4 years old when I bought it and it came with a 30 day warranty.
Even if a dealer does not offer a warranty there are companies out there where you can purchase a warranty on a used car, but they are not going to be cheap, and generally not worth it on a car of several years old.
It sounds like you are trying to purchase an older/cheaper car. Your best bet is to shop around and check any potential purchases out carefully while doing research on them before purchase.