Wikipedia: “The Karabiner Model 1931 (K31) is a magazine-fed, straight-pull bolt-action rifle. It was the standard issue rifle of the Swiss armed forces from 1933 until 1958, though examples remained in service into the 1970s. It has a 6-round removable magazine, and is chambered for the 7.5Ã—55mm Swiss (also known as Gewehrpatrone 1911, GP11, or unofficially 7.5Ã—55mm Schmidtâ€“Rubin), a cartridge with ballistic qualities similar to the 7.62Ã—51mm NATO/.308 Winchester cartridge.”
The video below, made by the Long Range Shooters of Utah, shows Ernest Jimenez setting a new long-range record for shots with an open sight rifle, firing at a 36-inch (.9144 m.) pink bison target posted 2,240 yards (2048.256 m.) away.
Mr. Jimenez was using an unmodified Swiss K31 rifle which he originally purchased for $99. You can buy one of these today for a bit over $300. (Classic Firearms has them.)
Mr. Jimenez scored 4 hits in the course of firing something like 83 shots. (Of course, a lot of his misses came awfully close.) On a completely windless day, he would likely have done better.
Personally, I think you could do as well, or better, using a 1903 Springfield.