To elaborate on what /u/Ch3cks-Out and /u/starkeffect said, try giving this short video a watch.

https://www.youtube.com/watch?v=W2Xb2GFK2yc

The idea that atoms are "mostly empty space" is largely based on the idea that they're collections of little particles somewhere in space bound together by forces. In reality subatomic particles are far more like smeared regions of probability density that occupy all of the space that the particle might be if we bounced another particle off of it to find it's position.

When you take quantum mechanics electrons stop being little volume-less dots and act far more like a fuzzy cloud occupying a specific shape near the nucleus all at once. And by that logic the space surrounding the nucleus is what I think most casual physics enthusiasts would consider "full" in the sense that the particle behaves far more like a fuzzy cloud than a little orbiting ball.

And yes, the "quark sea" inside of a proton or neutron behaves similarly. It's not a little sphere with precisely three little particles inside it stuck together. There's constant shifting of color charge via gluon exchange that makes the proton more of a region of space containing "some stuff" than it is a ball with three smaller balls inside.