Before I make any chambering recommendations I would want to know what is the purpose for the long-range shooting - probably Target, but could be Hunting. More importantly, what do you consider long-range? 400 yards, 600 yards, 1,000 yards?
To most hunters anything over 300 yards is too far away to be taking a shot. If you are a truely gifted marksman, you can certainly shoot that far, but generally do not have to do so to make your kill.
If you are a varmint hunter, then the ultra long-range shooting is brought to bear. I think the longest confirmed kill in 2003 was right at 2,800 yards. The .300 Win. Mag. set up cost something like $8,000 to custom build and the guy that built it owned his own machine shop. It takes pretty good shooting to nail a 10 inch tall by 4-5 inch wide critter at nearly 2 miles.
The NRA has Palma matches that shoot out to 1,000 yards with open sights. These competitors shoot incredibly small groups from long range using a 155 grain .308 bullet and no scope. That's shooting.
Alternatively, you could shoot highpower rifle (probably with a .308 or .223) out to 1,000, or "F" class at 1,000 and use a scope. Lots of alternatives to choose from.
You can use a rifle chambered in .308 for any or all of these long-range events (although event specific rifles are generally the norm - meaning a Palma rifle is not something you would use in a highpower or "F" Class match). Lots of choices.
I use a .308 now (but have a 6.5x284 on the way for F Class out to 1,000 yards). My 700-PSS shot 1/2 MOA right out of the box. It cost roughly $700, but then there is the scope, bi-pod and ammo to consider. All-in-all not cheap.
You could do it with a different and less expensive rifle, like the Savage 10-FP, and I suspect there are few "box" rifles that will equal these two in accuracy (maybe the Remington VSS or somebody's standard varmint rifle). If I'm wrong we'll both know better in a second or two.