As Kraig mentioned, you can put a very accurate CMP barrel on there and have a rifle with very acceptable accuracy.
Or, you could go the custom barrel route and have it totally tuned up. And yes, with the right cartridge and load, you can get 1/4 MOA out of these. The action itself is very smooth and there is a lot of "meat" in the action. It will hold very stiff when shooting. Shooting a cartridge that does not produce as much recoil as the 30-06 will probably even better your chances for super tight groups.
That said, it still depends on the shooter too. And, know that if you go just getting a CMP barrel on the rifle, or don't do all the things necessary to make your rifle with a custom barrel a better shooter, it's not going to produce super groups. You need to spend the money to have your gunsmith take the time to do all the things that can make these exceptional shooters. You'll spend a lot of money getting that last 1/2 MOA of accuracy. But, you would do that with any custom rifle you want to be super accurate. The difference with these and a Rem 700 (or clone) is that most gunsmiths are set up for 700's. And, a 700 has less issues when truing up an action and mating a barrel to it.