What evidence suggests that nuclear forces have a powerful force of attraction?

Many atoms have nuclei consisting of several protons packed into an extremely small volume. Each proton has a positive charge, and since positive charges repel each other, the theory of electromagnetism says that these nuclei must not exist in a stable state. The fact that they do indeed exist leads us to believe that there is a force holding the protons together which must be stronger than the electromagnetic force. We call this force the strong nuclear force.

In summary, the fact that protons are held together in nuclei despite their electric repulsion is evidence that there is a force (called the strong nuclear force) which is stronger than their electric repulsion.