A .338 Winchester Magnum is definitely a long range caliber in the hands of a long range shooter.
The problem is, long range shooting comes with lots of practice and experience, and a .338 is a tad more expensive in achieving this experience. However, don't let this stop you if it is what you want.
You will also need to find a place where you can shoot long distances.
A .308 can equal or beat a .338 at the range, where you have the advantage of known distances. However, if you are hunting, wind and range estimation is made less critical by heavy bullets going at magnum speeds.