Hit up your local pawn shops for a "donor" Savage action. A beat up hunting rifle is fine... all you want is the action and stock, chambering is irrelevant as you're getting a new barrel.
If you find one with a good barrel, you can sell it to reduce your "net cost" on the project. Maybe you can even find one with a decent optic, which you could also keep and use.
Figure about $200 net for a short action rifle with a .470 boltface.
There are somewhat better quality barrels available from Criterion and others, but given budget- buy a Shaw (I have one, it shoots 3/4 minute) for $200, here:
http://www.gunshack.com/barrels
Chambering can run from mild (.308) to wild, up to you and whether or not you reload. At only 600 yards, the .308 will perform very well and would be a solid choice.
That leaves $200 for an optic and rings/mount- which isn't much, but I think you could find something used that would deliver.
You'd need a few tools and a "go-gauge" to complete the build, but you could perhaps borrow them, or sell them after. Don't know if "tools" required to build the rifle would be considered part of the actual rifle cost.