Savage rifles are your best bet for accuracy under $500, but as others have said, you're not likely to build anything in that price range. The Accu-Trigger is the best factory trigger on the market, hands-down, which goes a long way toward accuracy.
You could "build" a TC Encore that will shoot out to 500 yards for a little more than that. I paid $600 for mine, used, with scope, in 7mm Mag and it shoots 3/4 MOA if I do my job, but that was a lucky find. Nice thing about the Encore is you can change calibers easily, it's light and short compared to bolts, and cheap for the accuracy you get. Downside is the break-action is hard to operate when prone and slower than a repeater regardless.
If you're really looking to build a rifle, starting with a Rem 700 action is the way to go, as they have the most variety of compatible parts, which usually means you can find more bargains. But $500 is still too low unless you get really lucky on some deals.