I don't know much about Russian law, but yes, AFAIK, Russian civilians can and do own some firearms. Lots of shotguns, some old rifles, and there MIGHT be a few who can own some target .22LR handguns, for competition practice?
I seem to recall that even us foreigners can go hunting in Russia, if we have the bucks to do so. I believe that Russian civilians are not allowed to own firearms chambered "in military cartridges"...which one would assume means 7.62x39 and x51R would be banned?? This apparently is where Russian civies have that 9mm big bore rifle for hunting bears and so on.
And yet, I've also seen on TV Russian civilians carrying AKM's and SKS too...so maybe they just "use" them without "owning" them? I dunno! Even their kids who join the "Spetnaz kids" (has a specific name that escapes me now) get to jump out of AN-2's and shoot AK-74's. Probably a bit like our Army Cadets from the 1970's and 80's.