I just watched a program on the History Channel called "Sniper, Behind the Crosshairs" (I think) and they were highlighting one Marine that was in a position in Iraq for about a week with his spotter... Many other stories I've read about the Viet Nam War had the snipers going quite a way behind enemy lines for their assignments...
The number of rounds were more than a "handful" ... and some of the distances required cartridges that would be accurate well past the normal 600-1,000 yards...but they seemed to do well with the .308..
Most city law enforcement counter sniper units are shooting under 200 yards...but due to potential barriers (glass, walls, cars, etc), the .308 is more decisive than the .223...