the answer to that question relies on three variables:
the terminal ballistics of the bullet
the tolerances of the weapon
the ability of the shooter
a 175 grain, FMJ boat tail, max power .308 load fired from an 22" barrel will have somewhere in the neighborhood of 1700 ft lbs of energy at 500 yards. that is plenty to take out a deer, but probably not enough to take a large bear reliably (just a guess, i dont know much about bears and other super game).
at 1000 yards, the same round has dropped to about 550 ft lbs. which many would argue is not enough to take even deer sized game.
also consider drop and wind effect. at 500 yards, the bullet has dropped about 50 inches. if there was a constant 10mph crosswind, it would be off by 22 inches or so.
as far as the saiga .308 is concerned, i saw a thread (maybe on a different forum, not sure) where a guy using handloads was able to get .5 MOA without any serious modifications. i dont know if that would be possible with factory ammo, but the gun is capable of being extremely accurate.
one more thing: its pretty hard to make field shots at those distances. before you take a shot like that, you need to make sure that you are capable of putting the round in the correct place to ensure a fast kill. if it isnt a fast kill, your long distance from the target will make it extremely difficult to track it.