Get an action with a "standard" bolt head, which is .473". This will include calibers such as .308 and all it's variants, such as .243, 7-08, .260, etc. are all common standard short action calibers. In long action, common calibers are the .30-06 and it's offspring such as the .270, .280 Rem, and .25-06. The .284 Win you are interested in also uses a .473 bolt head. There are many more, but these are some of the most common.
If your shooting will be limited to 600yds, I'd definitely consider the 6BR or one of its variations as one of the above posters suggested, especially if you are a handloader. Even if you aren't, very accurate factory loads are available, albeit not as easy to find as many other loads. 6BR is a very accurate, easy shooting round that is already a major player in 600 yd competitions. If you were to be shooting at 1000 yards, the .284 or the 6.5-284 would arguably be better choices. But you can't go wrong with the 6BR at 600. Less recoil, longer barrel life, and great accuracy. If you are seeking a factory rifle, I believe Savage offers a few models in this chambering.
If you just cannot accept the idea of shooting a 6mm for whatever reason, I'd suggest the 6.5 Creedmoor as another alternative that has good factory rifles and ammo easily available. It also uses a .473 bolt face if you want to build your own rifle.

Edit: 6BR also uses a .473 bolt face.