How does the currently accepted model of the nucleus provide evidence of the existence of the strong nuclear force?

We know the electromagnetic force exists, and that charges exert forces on each other. The current accepted model of the atomic nucleus is composed of a number of protons with charge + e and neutrons, with neutral charge. These positive charges exert forces on each other, encouraging the particles to accelerate away from one another. Since they do not, as the atomic nucleus is relatively stable, there must be some force counteracting the electromagnetic force.