I bought an old Russian 91/30.
There are ways to slightly improve the accuracy of them using basement(redneck) non permanent gunsmithing methods.
First, put a thin piece of plastic or metal under the screw area on the sear/trigger spring to drasticaly reduce the pull required to release the firing pin. Actually, I used a standard credit card thick piece of an expired ID card. It is the perfect thickness.
You can also modify a clothes pin spring to make a "two-stage" trigger and eliminate the slop inherent in the mosin trigger after cocking. I just finally got this figured out yesterday and installed

Took a lof of dicking around with a pair of pliers but it works awesome now!
The other thing, which the Finns actualy did on captured russian rifles to improve the accuracy was to place metal shims in the stock. Specifically under the top rear receiver screw, you will have to drill a hole through it so the screw can pass. It focuses the angle of the barrel into the stock. Depending on the condition of your stock, this will cause a bedding effect if your bands are in good condition. Alternatively you can use flat pieces of cork at equal lengths to bed your stock without actually permanently altering it. Some people try to free float the barrel but this is pointless in a Mosin and causes more pain than it's worth.
Another is that some Mosins shoot
VERY high. Partly becuase they were designed to. You can buy a piece of 12 guage insulated electrical wire, strip off the insulation and slip it over the front post. Keep shooting 4 round groups at 100m and trim down the post a smidge at a time until your groups are dead center POA. Just marker the insulation black after, I found it easier to pick up the sight image leaving the tip white actually so that was a bonus! This was an issue with mine as the front post had been visibly hacked very short. I was shooting more than 2 feet high at 100m!
The last and most important, get your barrel recrowned. You would not believe the improvement in accuracy you can achieve with this simple step.
All this for nothing more than $50

There is a wealth of info on this subject on the net. Just stay away from the free floating barrel nonsense.