Is the city liable for damage because of a blocked sewer line?

It depends on whose property it's on. If it's past your property line, it's your problem; if it's on the city's property, it's their problem (also theirs if they have a utilities easement or legal right-of-way on your property that pertains to the problem. *I am not a lawyer, this is not legal advice and you should seek counsel if you intend to take any action*