If you are looking to shoot tiny groups, you will probably be disappointed with the Mosin Nagant, but they are normally quite capable of the accuracy needed for the role for which they were designed, hitting a man-sized target out to any range where a rifleman can pick out and target an individual enemy soldier.
You can improve things a bit by using good ammo, and by doing the things to the rifle the Soviets did when getting them ready for competition. Great discussion and notes from a Soviet era book on the subject here:
http://forums.gunboards.com/showthread.php?294174-Yur-yev-quot-Sport-Shooting-quot-Book
The Reader's Digest condensed version:
Make sure the barrel does not contact the end cap:
Make sure the action fits snugly in the stock with full contact here:
Make sure the action does not contact the stock here:
Make sure the action does not contact the magazine:
Float the barrel and handguard, then wrap the barrel in cloth or felt even with the front barrel band:
I understand the Finns used cork, but I haven't seen a description of what/where exactly they put it, maybe Josh can provide that info.
Not a Mosin, but here is an example of what you can do with an old rifle with iron sights, shooting my 1917 Mauser at a standard steel IPSC silhouette target, laser ranged at 470 yards:
https://www.youtube.com/watch?v=xWOePGiaBcM
I have done the same thing with a mosin, just didn't have video running.