Sorry, I should have specified felon rather than simply having a criminal record. Generally speaking any crime that carries a possible prison sentence over a year, though you're right that it can vary by state etc. And blackpowder guns and a few others aren't legally considered "firearms" by the ATF, so a felon can own them.
Which is why in The Dukes of Hazzard, Bo & Luke carried compound bows (with dynamite tied to the arrows!). Because they were moonshiners (a federal felony) they couldn't have guns.
That's not true at all... It depends on the crime, it depends on the state, and it depends on the firearm.