Elevator wires are engineered and made of steel and other composite materials. Also, instead of a single wire, they are several strands of different sizes twisted together. A typical cable or rope can have over 150 strands of wire engineered to be strong, flexible and long-lasting.
Almost all pulley elevators have multiple cables -- four to eight in total. Even if one cable breaks, the remaining cables will hold the elevator car up. In fact, usually just one cable is enough.