In Russia it sounds very XIX century, a name from classic literature. Quite harsh. V-r-v-r. But it seems to return these last 5 years or so: I hear more and more about little girls named Varya. Good name for singer or fashion designer, still pretty much an out of line, drawing attention kind of name.
I used to kind of like this name, and the nickname Varya, but now I dislike it as much as the English name Barbara. Varvara is as dated in Russia as Barbara is in the English-speaking world, though I've heard it's starting to be used a little bit more in recent years. It's just too heavy and old-fashioned for me.
Don't forget that the word "barbaros" (barbarian) originally was not necessarily derogatory. (It just meant a foreigner (one of those people who go "bar-bar-bar-bar").) (In Ancient Greek the letter "beta" had a b sound, not a v sound as in Modern Greek.)
It means barbarian, so that's not really the nicest meaning, but it's a pretty name. I like it better than the English Barbara because I feel like the b's in that name are too harsh.
Varvara Petrovna Stavrogina is the name of a character in Fyodor Dostoevsky's book "Demons." She is a wealthy older lady whose son is the primary character in the novel.
Princess Varvara is a one-chapter character in L.M. Montgomery's book Magic for Marigold. Non-fictional bearers of this name were : Princess Varvara Nikolaevna Gagarine (1762–1802)