Those criteria don't sound very hard to fulfill at all, except for maybe the price. You might be able to get those groupings with a decent pistol, and a decent rifle will do this even easier. You definitely don't need a bipod, and shouldn't even need a rest for the gun.
You definitely don't need any sort of scope for just 28 feet. If you really want magnification, I'd say 2x at most...3x is pushing it.
If you're just plinking, it's up to you whether you get a bolt action or not. It's your preference, really.
The problem I see is getting a cheap gun with consistency. That is usually the realm of guns over that price range, but still, a stock gun of any decent quality should do the trick, I would think.