I'm not Russian but I've lived most of my life in the USA and still have a foreign accent, and I hate conversations about my country of origin, so I can relate.
I've learned that simply answering truthfully leads to a predictable sequence of annoying follow-up questions I don't want to answer for the 100th time. I've also found that simply refusing to answer is rude to the point of killing any further conversationimpacting future relationship with the asker (this is bad in a professional setting).
No answer is perfect but I've found two patterns that I've tended to rely on over the years:
"I grew up in"I'm from Los Angeles." Now the ball is in their court. Nice people will change subject or ask about LA (what neighborhood, do you know any good restaurants, etc). Rude people will press with something like, "No, where are you really from?" - this person is nosy, so let's have some fun: counter with the non-answer "I moved there when I was very young." ... At this point the persistent ones will switch to, "No, I mean, where is the accent from?" - easy answer, "Oh, the accent is because my parents spoke (language) at home... " still giving no conclusive info since many countries speak that language, while also starting to challenge their simplistic stereotypes.
"I'll give you three chances to guess it." Most people will NOT get it right on the first try (i.e. in your case they may pick any other slavic or former USSR country as their initial guess), and if they do, just ask for two more guesses. Here's the trick: start a conversation about each of their guesses. Oh, country X? Beautiful place, nice beaches, bad food. Have you been there? Do you know anyone else from there? This will stretch the conversation long enough that by the time we get to my actual country, it's time to move on. For me this works because I don't mind disclosing my country of origin, I just don't want to have a whole conversation about it. Only you can decide if this is suitable for you or not.