I'm linking a previous comment that summarised the conclusions of a seminar on the pragmatics of studying Russian history that I attended a month ago.

TLDR, it's bad. There are work-arounds depending on your area of research but the overall picture is quite bleak. I don't know where you're from, but the consensus seemed to be that obtaining a research visa if you're from a "hostile country" (in my case, the UK) is an impossibility, even if you felt it was safe and anybody was willing to fund you. For Soviet material, your best bet will be checking out the archives of neighbouring or ex-Soviet states (Georgia, Finland, Kazakhstan, Lithuania, Latvia, Estonia etc) to see if there's anything relevant there. A lot of those places were left by the remains of the Soviet bureaucracy in a hurry in 1991 and they didn't have time to shred everything.