So basically you are describing why objects are solid while saying this proves they're not. You even picked the wrong subatomic particle to make your case.
Really what it comes down to is objects get close enough to be considered touching. When we get down to angstrom scale (10^-10m) there may be small gaps, but it really depends on how you think about it.
Radicals by their very definition are atoms with unpaired electrons. Therefore only ions can be devoid of electrons, cations to be exact (anions have a negative charge cations have positive charge). You are of course not going to strip off all of the electrons this with most elements. The easiest would have to of course be hydrogen. The hydrogen carbocation [carbo cat ion] ,hydroxide, itself although thought of as alone by the uninitiated most often has to be stabilized by a solvent. I at the moment forget if you can get it in the gas phase.
Does it occur to you that electrons are in motion almost constantly. In fact you can even polarize atoms, ie slosh the electrons over to one side of the atom. The nucleus is positive and is more static, although recent evidence points to the nucleus possibly being a wave as well! Typically we think of nuclear repulsion rather than electron repulsion.