Elevator cables are highly engineered and made of steel with other composites. Also, they are not single wires but several strands of various sizes wrapped together. A typical cable or rope can have over 150 strands of wire precisely designed to be strong, flexible, and give long service.
Almost all pulley elevators have multiple cables -- between four and eight total. Even if one cable snapped, the remaining cables would hold the elevator car up. In fact, just one cable is usually enough.