The new Intel Pentium 20th anniversary CPU is pretty loving badass. If you're willing to overclock it to just about the max it'll run games no problem. Unless you're playing a CPU heavy game then it's stuff but it keeps up with mid range gpu's pretty good. It also doesn't put off a ton of heat either so a cheapy aftermarket cooler should work. I think for it and a decent board you're pushing like $150-$200. You'll need a discrete GPU but you should be able to put together a decent computer for around 400-500ish
This gives me memories of an old Pentium 4 Dimension XPS gaming PC I used to have, except it'd have sort of loud fans. The PC had it stock clocked 3.6GHz and the old high-end Pentiums already put out too much heat, so that is why it came with loud fans.
Good times, except when the PSU would burn out and the fans would go full blast (motherboard/PSU design flaw, some other Dimension PC models had this flaw)