I would second the Mosin-Nagant as an alternative to the selections originally posted. You can find good examples of the 91/30 for under $100 and the 7,62x54R has ballistics close to that of the .308 or 30.06.
I have an Izhevsk 91/30 dated from 1931 and the iron sights with Czech silvertip 147gr ammo allows for 3" groups at 100 yards. Mine will shoot 6-8" groups at 200 yds which was the Soviet standard for the 91/30 in the hands of Russian conscripts during the "Great Patriotic War". That is easily "minute of Nazi" at that distance.
Milsurp ammo is still inexpensive and there is a lot of it available. You can also get modern (reloadable) ammo from Wolf, Winchester and others in both FMJ for target shooting and soft point for hunting.
Scoping a Mosin is also one of those things that purists will scream and yell over but there were sniper versions of the 91/30 that used a small scope. In the hands of a good sniper like the Russian sniper Vasiiy Zaytsiev of WWII fame, it's very accurate.