So now we know that the strong force binds quarks together because quarks have color charge. But that still does not explain what holds the nucleus together, since positive protons repel each other with electromagnetic force, and protons and neutrons are color-neutral.

So what holds the nucleus together? Huh?

The answer is that, in short, they don't call it the strong force for nothing. The strong force between the quarks in one proton and the quarks in another proton is strong enough to overwhelm the repulsive electromagnetic force.

This is called the residual strong interaction, and it is what "glues" the nucleus together.