The nuclei of most atoms contain multiple protons. Each proton has a positive charge. If objects that have like charges repel each other, why do the protons in the nuclei of most atoms not fly apart?

The charges of protons and neutrons cancel each other out.

Electrons outside the nucleus push on protons.

A strong nuclear force between an atom's protons and neutrons holds together the atom's nucleus.

Protons experience a more powerful attraction to the negatively charged particles in an atom's nucleus.

To answer this we must consider the four fundamental forces of nature. These forces are the gravity, the electromagnetic force, the strong nuclear force and the weak nuclear force. The gravitational force depends on the mass of the object thus it only has a very small contribution to the nucleus. The electromagnetic force exists inside and outside the nucleus. It is the cause why unlike charges attract and like charges repel. This is what the other forces should overcome to keep the nucleus intact. The weak nuclear force has a stronger force than gravity but is weaker than the electromagnetic force. It is the last force (the strong nuclear force) that overcomes the electromagnetic force. The strong nuclear force pulls together the protons and the neutrons in the nucleus preventing the repulsion of like charges.