Original cartouched and papered 1903A4 snipers run from $3-$6K of late.
I found a 1903A4 that had been Bubba'ized into a deer rifle, and spent several years restoring it once I realized it had the proper offset serial number on the receiver. Even a restoration can cost a buttload of money. I purposely avoided adding the stock cartouches on my own 1903A4 restoration, and I've electropenciled the restoration date and my name on the bottom of the barrel so any future appraisers after I've taken my dirt nap will know the real story.
A true 1903A4 has the offset serial numbers on the front receiver ring, no front sights or evidence of a front sight having been installed, a properly bent and clearanced bolt, and may come with either 4-groove or 2-groove barrel. The stock may be either the S-type "scant" stock with rudimentary pistol grip, or the 1903A1's C-type stock with full pistol grip.
Accuracy with a 1903 variant is best determined by the condition of the specimen you're working with. If it looks like it's been dragged to hell and back, with a cleaning rod-eroded bore at the muzzle, or pitting inside the barrel that's more pronounced than the rifling, then of course it won't shoot well. Problem is, there aren't that many examples of a pristine or even genuinely well-kept 1903 Springfield out there, compared to all those nice Swedish M96 and M38 Mausers, or Swiss M1911 and K31 Schmidt-Rubins, which really didn't see much wartime use or hard service.
Obviously, a well-maintained 1903A4 sniper with either M73 or M73B1 scope at 2.5x will give excellent accuracy. Mine goes about 3/4 MOA, and it sports a NOS 2-groove barrel. However, there was something to the old WWI adage about the Germans bringing a hunting rifle to the fight, the British bringing a fighting rifle, and the Americans bringing a target rifle. This October 1918 Springfield M1903, later upgraded to M1903A1 configuration, is every bit as accurate as my Swedish Mausers or Swiss M1911 Schmidt-Rubin. So accurate, in fact, that I was asked not to bring it to any more John C. Garand matches, so that the guys with the M1 Garands had a chance to take home a few trophies.