Thousand yards huh? Weapon aside, can *you* make that shot? It's pointless to have such a weapon if you can't.
A Savage 30-ought (30-06) is cheap enough, but a .270 has better sectional density than a 30-06, so is a bit more lethal. Neither are really thousand yarders unless you're skilled.
Ammo starts to matter at that range, and a spearpoint has better aerodynamics (but more likely to jam if you have a semi like a Ruger 10/22, but as I already said, that has other problems, like range), so a bolt would be helpful.
A .22 is sure an unlikely killshot of a medium-large mammal, placement would matter. Bigger rounds are a better bet, bigger rounds are also necessary to get a thousand yards.
I'd decrease my range and go with a .45 carbine to plink a large 'mansize' mammal at distance. Followed by a .308 @1000, followed by a .270 or 30-06 @<1000.
In sum:
.308 is your best bet, and can be had under a thousand, but you'll want good optics. You'll need a prone shot, and a stable barrel (sandbags). 7MM probably don't come less than a grand :), or as I said, get closer with a .45 carbine :)