There are many rifles which will get the job done to 1000 yards. Three of these are based on a Remington 700, two Accuracy International, one M1A.
RiflesBasic / Economy. On the basic end are factory rifles like the Savage 10FP and Remington 700 PSS. These are sporter rifles with heavy barrels and target-style stocks. Their triggers need adjustment and/or tuning for a nice 2-4# break, and they need a scope base installed. Most of these rifles will shoot better than 1 MOA with some factory match ammunition, but a few will be lemons. The most likely problems will be poor ergonomic fit due to stock design. All the problems can be fixed with money spent replacing the stock, adding a detachable box magazine, adding a good trigger, replacing the barrel with a premium match barrel, getting the action trued, etc. You should be able to get a 10FP or 700 PSS for under $700, then add a base and rings, a scope and a bipod.
Custom. There are many gunsmiths who build precision long-range rifles. The vast majority of these are based on the Remington 700 action, with real performance improvements done to them, and some frills. Some are based on better actions such as the Surgeon, which has a Picatinny rail machined right into the top of the receiver. The builders I recommend are GA Precision, KMW Precision, and Surgeon Rifles. These guys all have impeccable records and deliver a lot of well-built rifles for practical long-range shooting.
High-end Factory. The other option is a high-end factory rifle. These are generally rifles designed from the ground up for military sniper duty, and sold on the commercial market as well. The most notable are the Arctic Warfare line from Accuracy International (U.K.), and the TRG line from Sako (Finland). The quality and accuracy of these rifles are basically the same as the custom rifles mentioned, but they are factory pieces and generally have a "system" of accessories to go along with them, as well as having a logistics trail already. Replacement parts are stocked items, not custom one-offs. $3000-5500.
Most new long-range rifle shooters will be able to shoot a long time and learn a lot using the basic Remington 700 or Savage 10FP, if they get the rifle to fit them and mount quality optics on it. For the shooter that never wants to question if his equipment has let him down, the AI AW, Sako TRG, or high-end custom rifle from GAP, KMW, or Surgeon is the way to go. Putting more money in a better rifle is an investment that will pay off down the line.
AmmunitionThere are basically two choices. You can buy factory ammunition, or you can shoot hand-loads. In some calibers, there are no really good factory loads appropriate for long-range shooting, so you effectively have to reload.
The best sources of factory ammunition for long-range shooting are: the Gold Medal Match line from Federal, and Black Hills Ammunition. .308 Winchester has the most factory match-grade ammunition available, so if ammo supply will be a problem and you cannot reload, .308 is definitely the caliber to choose.
Here is a cautionary note for .308 Winchester ammunition selection for long-range use. The .308 can deliver good thousand-yard performance, but only with careful selection of ammunition, specifically the bullet used and the muzzle velocity. Those two together must be sufficient to keep the bullet solidly supersonic when shot through the current ambient atmospheric conditions. Specifically, standard 150gr FMJ and even some match bullets have BC values too low to reliably stay supersonic at this range. In denser atmospheric conditions, they may produce poor accuracy or even strike the target sideways. The 168gr SMK bullet is known for producing variable results at 1000 yards. There are better choices for .308 bullet selection, so just don't use the 168gr SMK. The Sierra MatchKing bullets heavier than 175gr will work. Any of the Berger VLD bullets heavier than 168gr should work. The best choice is the 155gr Lapua Scenar, if it shoots accurately from your barrel. This bullet has a BC value as high as the 175gr SMK, but can be shot 100-200fps faster from most barrels.
Reloading is financially compelling. Where factory match might be $18 a box of 20, by saving your brass and reloading, you can save 50-75% of the cost. That translates into more shooting for less money. The other benefit is that you can tailor a load for your specific rifle that suits your needs.
There are some excellent long-range calibers for which factory match-grade long-range ammunition is not available, or is available only in very small quantities or for a very high price. These include all of the calibers discussed above except for .308, 300 WM, and .338 Lapua. If you own a rifle in a caliber for which good long-range ammunition is not available, you have committed to quite a bit of load development and reloading time.
When given a choice in components, I recommend choosing Lapua cases (brass) if they are available in that caliber. After Lapua, try Norma, then Winchester, then whatever you can get. Bullet selection was covered before, but some good candidates are bullets from Lapua, Berger, and Sierra. For powder type, I prefer to use powers from Hodgdon's "Extreme" line, including H4831SC, Varget, Retumbo. There is always some experimentation involved in developing "THE" load for your long-range rifle.