In your spatial visualisation a proton consist of two up and one down quark spinning around at extremely high speed forming a type of 'cloud' that can be perceived as being the proton.

A neutron is a similar arrangement but with two down quarks and one up quark.

A proton is around 1x10^-15m whereas a quark is at largest 1x10^-18m so in crude terms there are 3 items (at least) 1000 times smaller than a proton whizzing about inside the proton so I would say based on this that a proton is around 99.97% empty space. Neutrons appear to be only very slightly more massive so can effectively considered to be equal in 'size' to a proton.

But this does not take into account the soup of gluons and antiquarks in this arrangement, or that the quark may even in fact be a true point particle in that it has mass but occupies a nonzero amount of space.

They're not like ever decreasing billiard balls but more akin to the leading edge of a helicopter blade, imagine that the helicopter spins its blades at the speed of light forming a seemingly solid disc with what would appear to be a solid 'edge' however, the blades would be imperceptible individually. Again a crude analogy but it comes closer to approaching a visual representation.