I have a Finnish version of the 91/30 with a .308 bore. It shoots so well it's scary. I have bad eyes and can get it under 4" at 300yds with iron's! With a little bedding and some trigger work, they can be made to shoot very well. I have a Huber Concepts trigger in it as well, and it's a huge improvement over stock.
If you have a rough barrel, fire lapping is an excellent way to improve the accuracy. Using valve lapping compound on cast bullets is the best way to fire lap them with the oversize bore, as Tubb's final finish only has .308 bullets, which will never hit the grooves.
I used .313 cast bullets (with soft lead to allow the compound to impregnate in them, basically straight wheel weights) with valve compound on my Russian 91/30 that has a .3115 bore and it improved things greatly. All you need to do is put some lapping compound on a flat and hard surface smooth it out so it's not in a big lump, lay a cast bullet in it, and with another hard flat object (I used a piece of aluminum plate) roll the bullet between them in the compound, but don't press so hard that it will size the bullet down, which is easier than you think. This will impregnate the bullet with the grit. I then lubed with Lee's Alox/beeswax stick lube in the grease grooves, and loaded as normal over a light charge suitable for cast bullets. You can even roll it in the compound again after lubing but before loading, it's all up to you. After firing about 30-40 of these, the bore was much smoother than before and cleaned much easier than in the past, and was more accurate as well.
The Mosin Nagant gets a bad rap that is undeserved for being inaccurate. Many of the guns that are inaccurate just have abused barrels or are used with loads that are not suited to the particular rifle. I have improved the accuracy of most all my Mosins (I have 5) with just a bit of handload tuning. And my Fin will go under an inch @100yds with irons, and under 4" @300yds, and I have bad eyes. If one of the talented prone shooters from the forum were to shoot my rifle, I bet it would be very accurate, and would likely be able to shoot MOA or better at 600 and even 1000yds.
The Mosin is a great rifle with a very strong action. And the 7.62X54R isw an awesome round as well. It's more powerful than the .308, and nearly on par with a .30-06. It is capable of great accuracy if tuned well, and will kill anything in the mainland US if used for hunting.