While 338 is likely a better caliber for 1000 yards and beyond, for most folks, myself included, the cost of shooting it even with reloads is prohibitive, not to mention the cost of the rifle. I believe a low-priced 338 Lapua magnum in the Savage or Remington flavor will run around $2,000...a used one will be several hundred less, but much like a sports car or 4-wheel drive truck, you never know how it's been driven.
A high-end production 308 Win, on the other hand, will run you about $1200 for a good long range rifle and 50 cents per round to reload...maybe slightly less if you use once-fired Lake City brass. I would recommend pulled surplus military m118LR bullets in lieu of new matchkings, but the price of mil surp pulled bullets is now almost as much as new 175 gr matchkings.
I have engaged and hit targets out to 1100 yards with a Rem 700 5r mil spec rifle in 308 Winchester. These were not "hot" reloads, and departed the muzzle around 2620 fps, much like the $2 Federal Gold Medal Match rounds, but at one fourth the cost.
Another option is the venerable 30-06. It might be a little harder to find this caliber in a target/long range rifle, but if you look hard enough, you may be successful. It has a slight advantage in muzzle velocity over the 308 Win. Whether that advantage is worth the additional ammunition cost and recoil is a personal preference.
Other options like the 6.5mm rifles and even the 6mm/.243 will also get you out to 1000 yards, but as stated before, ammo availability is an issue unless you reload; in all of these calibers, that is true at the moment.
Whatever you choose, you should consider your primary purpose for the rifle, then purchase based on that. You can hunt with a target rifle, it just weighs a bit more and makes you tired of carrying it.
If you are serious about long range shooting, you also need to consider the price of a quality optic to go on top. For a target rifle of good/high quality, I would expect to spend a minimum of $500 on a scope. If your budget allows, more $$ will buy you clearer glass with better wind/elev adjustment repeatability.
Choose wisely.