What is Solid about Matter?

First of all, let me say that your question is exceptionally well framed and introduces several concepts in the most interesting way. Well said! (Well asked?)

If you stop for a moment and consider your common everyday hydrogen atom, you realize that it is mostly empty space. As the atomic (not nuclear) structure of the other elements is not all that fundamentally different from hydrogen, you can safely assume that all ordinary matter is far from being "solid" in any sense of the word.

Just to give a clear idea just how empty an ordinary hydrogen atom really is, place a marble at the center of the 50 yard line in the middle of a major football stadium. At these scales that marble, a single proton, is just about the right size to represent your hydrogen atom's nucleus.

Then, take a speck of pepper - your hydrogen atom's only electron - and imagine it in a fuzzy, rather spherical orbital which neatly encloses the stadium itself. Now imagine all the other elements at this scale and you can probably guess that they, too, consist mostly of empty space. Even U238, with its massive nucleus and surrounding fog of electrons, is mostly empty space.

When you strike a nail with a hammer, the force of the impact is felt only by the outermost electron orbitals in each piece. Meanwhile the nuclei in all those iron atoms remain locked inside their electronic cocoons and feel nary a thing - remarkably like King George, actually, whose little bubble isolates him from life in the Real World.

Scaling things to the other extreme, you now imagine yourself being able to see spacetime itself at Planck-Length scales. Not even here can anything be "solid" in in terms of being everywhere connected and continuous. At this scale - and if you could "freeze" everything in order to take a closer look - you'd discover that spacetime is essentially "porous" (for lack of a better word), like some kind of exotic sponge. But unlike a sponge, the 'pores' in spacetime are not only empty, but within them spacetime itself ceases to exist. The pores are, in a sense, "filled" with nothingness. (Don't think for a minute that I find this possible to imagine. It's more a mathematical abstraction than a visual one.)

Consequently, as to your question of there being any truly "solid" matter? I'd say the chances for that are probably about the same as a snowball's chance in Hell.