It means that the live on a street that is directly connected to Fifth Avenue. For example. You're driving down 5th Avenue and on your left is a road called Elm Street. As you keep going there is another road called Lane Drive. Both of those streets would be a road they could possibly live off (because they are streets OFF 5th Avenue) Now if the person said "I live off 5th Avenue" that would basically mean their house is located on 5th Avenue.

They live in a street off Fifth Avenue. In this context 'off' stand for 'away from'. Here we go: They live in a street, (located) away from Fifth Avenue.

@MoiraWendy Miller, understand, thx.

I live in a street off York Road. This means my little road connects to York Road. I give directions by saying, "It's off York Road which is off Clarkson which is off Lynn Road." They find it !!!!

Yes, Brian. You have to educate me on the Shanghai system, though I have spent time in Shanghai. I find that in some areas the grid reference works well, e.g. in the Bund area, and along Huai Hai Zhong Lu (which was the old Avenue Joffre in the French Concession), but in some instances, where there are many sub-streets called alleys (比如，第3里第4弄), the alleys sometimes run like a labyrinth. By the way, though I am not a supporter of colonialism, I wish they had kept the old French names, rue Molière, etc.