If you seriously want to try 1000 yard shooting then you will need to fork out some serious money to do it and have any chance being effective at it.

Don't expect to just go out and buy an off the shelf gun and hit anything at 1000 yards. The bench rest guns are highly specialized and even the rifles used to hunt at 1000 yards are incredibly specialized. Heavy stocks, long and heavy super match grade barrels (26" and longer), match grade triggers such as Jewel. You can easily spend a few to several thousand for the gun, another grand for the scope, another grand for a spotting scope, etc... etc...

You need to have specialized reloading equipment so you can wring out the most accuraty from your hand loads.

You will need to spend countless hours at the bench firing countless rounds of ammunition in order to get good enough for this.

For paper punching a 308 will work fine IF you reload heavy bullets (175 grains +) to at least 2600 fps so that you can keep the bullets super sonic at 1000 yards. If you want to hunt big game at 1000 yards then you would be muh better off with something along the lines of a .338 Lapua, .338 RUM, etc... in order to have sufficent energy levels at 1000 yards.