.223 Rem will work fine for the distances you're interested in shooting. Plus it's cheaper to feed than a .308 Win whether you reload or shoot factory stuff.
Whatever you decide on, get a barrel with a 1:9 or 1:8 twist to shoot longer 68 - 77 grain bullets.
The Savage Model 12 Varmint Series or Model 10 Law Enforcement series come chambered in .223 Rem with 1:9 twists (some with 1:7 twists). They're pretty accurate out of the box, reasonably priced, and I like the adjustable accutrigger.
There's plenty of other choices: Remington, Tikka, Sako, Ruger, Browning, etc. If you're going to have a smith build one and don't already have a donor action, consider going with a custom action.
Savage makes lighter weight rifles than the Models 10 & 12. If you're wanting a carry rifle, consider one of those. Leave enough money set aside for a decent scope.