You can use .308 dia. bullets BUT I would have the barrel slugged first. Some of the mosins had a bore diameter up to .314. I shoot 125 and 150gr nosler ballistic tips out of mine. I use H335 and H4895 powder.
I tried some 308 bullets and the accuracy was terrible.
The bullets to use are the ones for 303 British (Enfields). But to be safe you should probably slug the barrel first. There is little chance that it is 308 but who knows. A too small bullet in a too big bore is only an accuracy issue but a too big bullet in a too small bore is dangerously deadly.
Bores on these Mosin Nagants vary all over the place but mostly are up to 0.314. Only the 303 Brit bullets (actually measure 0.311, I believe) can get you safely close to the average Mosin Nagant bore size.