I just bought a refurbished wartime (1944) 91/30 at Big-5 Sporting Goods for $75. Bore condition is OK; a lot of wear but not much pitting. I've been to the range a couple times now and the best I've been able to shoot with it for accuracy has been 2.5MOA groups (minute-of-angle- about 1 inch at 100 yards, 2 inches at 200 yards, etc.) with the occasional flyer. I haven't tried using shims to bed the barrel yet to see if I can eliminate the flyers, but that might work.
The average Mosin Nagant will probably do 3MOA or better with the right ammo. I personally wouldn't put a scope on a 3MOA rifle, unless you have eye problems. With some practice 3MOA is not overly difficult with open sights (from a rest, of course).
As far as range, 7.62x54R is pretty comparable to the 30-06, which has the ballistics to go over 1000 yards. So theoretically, if you got a good Mosin Nagant that could shoot 2MOA, you could hit a 20 inch circle at 1000 yards (definitely scope time! I doubt you could see a 20 inch circle at that distance). I say theoretically because there are so many variables at the longer ranges. You have to be a very good shot to think about shooting that far.
I remember reading a book about Carlos Hathcock, the top sniper during the Vietnam war. The enemy snipers used Mosin Nagant sniper rifles and were frighteningly lethal at impressive distances. I'm sure they were able to hand pick the most accurate rifle, and undoubtedly one that has been accurized could be very accurate.
I know there are several here that are more knowledgeable than I am about MNs. This is just my two cents worth.
Edit: sorry Longrifleman, I didn't see your post when I posted, was a little redundant.
Also- it turned out my 91/30 is a re-arsenaled ex-sniper, with plugged and welded screw holes. It doesn't seem to be much more accurate than average, but for $75 I did get a good deal!