Are real handguns even available for the average citizen there? I know that some retired .gov officials and military officers can get licenses to own actual handguns, but I thought the average Russian had to settle for owning & carrying one of those rubber bullet firing handguns.
Depends on your definition of average citizen.
Nominally, handguns are banned. With a couple of exceptions.
You can register as a competition shooter, pay a lot of money, invest a lot of time, and buy a very overpriced pistol to be kept at the shooting club.
You can also be awarded a handgun in the same way you would get a medal elsewhere, for some kind of merit, like bravery, or many years of exemplary service. This is how those retired military and government types get them. I am also told that a donation of a certain amount of money or equipment to your local police station can be see as quite a civic achievement worthy of such an award. These guns cannot be inherited, sold, or loaned.
There is also a way to get an antique pistol as a collector. That requires a different license with a separate set of hoops. These guns can't be carried. I don't remember what the exact definition of an antique is. Also, there is no ammunition sold for them.
Other than that, you get a license to carry "weapons of limited damage", which are the rubber bullet shooting contraptions affectionately know as "condom slingers".