view that an atom mostly consist of empty space stems from the old times when Bohr's atomic model (as a miniature planetary system in which electrons surround the nucleus) was the best picture of what an atom is like.
But there are no electron particles moving around an atom. One cannot view the electrons as little balls moving inside a molecule and somehow avoiding falling into a nucleus. Such a configuration would be unstable. The nuclei would attract little charged balls until they fall into them.
But it is very well understood why atoms are stable - the ground state is a delocalized stationary state of the electrons in an atom, a state living indefinitely (unless the nucleus decays). In terms of quantum field theory, the space is filled by the electron field. The resulting electron density can be calculated by quantum mechanics. Indeed, this is one of the outputs chemists are interested in when they use quantum chemistry packages like GAMESS.
Electrons behaving as particles (in the sense of being localized at approximately one place) only exist in situations where one may consider the field theory in the limit of geometric optics (cf. photons as particles of light), so that one can speak meaningfully of their paths. See, e.g.,
Maloff, I.G. and Epstein, D.W., theory of electron gun, Proc. Inst. Radio Eng. 22 (2006), 1386-1411.
which presents a phenomenological view,
Jagannathan, R. and Simon, R. and Sudarshan, N., quantum theory of magnetic lens, Phys. Lett. A 134 (1989), 457-464.
which derives geometrical electron optics from the Dirac equation, or the book
P.W. Hawkes and E. Kasper, Principles of Electron Optics, Vol. 2: Applied Geometrical Optics, Elsevier, 1989.
which contains engineering details for electron beams.
Geometric optics is an essentially macroscopic view not applicable inside atoms or small molecules. To measure the position of a single electron, you need to make it reach a localized detector such as a Geiger counter, thereby localizing it. But it is impossible to make a measurement of an electron bound in a molecule. What one can measure there is only the charge distribution.
There is no empty space around a nucleus, as in Bohr's superseded model. The electrons make up a tiny proportion of the mass of an atom, while the nucleus makes up the rest. The nucleus makes up a tiny proportion of the space occupied by an atom, while the electrons make up the rest.
According to quantum electrodynamics, the space is filled by an electron field around the nucleus which neutralizes its charge and fills the space defining the atom size. What is displayed by a field ion microscope is the boundary of this field. But this boundary is not perfectly defined but a bit fuzzy, more like the surface of a piece of fur or of a cloud.
The electrons are therefore rather like a very low-density glue-like viscous fluid surrounding the nuclei and making up the spatial extent of the atom, transparent for neutrons but not for other electrons. Chemists draw the shape of these fluid clouds (more precisely, the electron density) as orbitals. Electrons show up as particles only under particular circumstances; e.g., in detectors such as Geiger counters.
The picture of an atom being mostly empty stems from the childhood of atomic structure analysis, where most of the atom's extension was found to be transparent for alpha rays, and the early models explained that by pointlike nuclei and electrons.
Similarly the picture of a proton or neutron being essentially empty apart from three quarks embedded in it arises because deep inelastic scattering shows that protons are essentially transparent for very energetic electrons, except when the latter meet an almost pointlike quark.
But both pictures are quite limited: We don't think glass doesn't occupy space because it is transparent for light, or that only the bones of our bodies occupy space because the remainder is transparent for X-rays. So why should we think of the electronic fluid surrounding nuclei not to occupy space simply because it is transparent to alpha rays, or of the meson and gluon fluid in which the quarks are embedded not to occupy space simply because it is transparent to fast leptons?
Glass is hard because it is occupied by a matter field that resists other matter (though not photons). Atoms are even harder because it is occupied by a matter field that resists other matter (though not alpha rays). Protons and neutrons are even harder because they are occupied by a matter field that resists other matter (though not fast leptons).
What we touch is an effective field whose extension is created by the electrons, and whose mass is created by the nuclei (or, on an even deeper level, by constituent quarks).
Indeed, most of the mass in ordinary matter is due to the strong interaction, generated dynamically through dynamical symmetry breaking. This results in constituent quark masses. These approximately add up to proton and neutron masses, and from these to the masses of atoms and molecules, and finally of the solids and fluids that make up our everyday world. The deviations are due to the fact that mass and energy are inter-convertible to some extent, and that the binding energy takes away a bit from particles bound together.
In macroscopic neutral matter, the effective fields are one mass field for each participating chemical substance, a stress field, a momentum field, an energy field. and their conjugate thermodynamic fields.
If two atoms or molecules touch, the volumes occupied by their electron fields touch, and repel each other, while at a slightly (but not much) larger distance there is a slight attraction, the van der Waals attraction, responsible for the formation of liquids. Thus touching is a real effect. The nuclei don't touch each other but the atoms and molecules do.
More precisely, the residual force between electrons bound in two different atoms whose nuclei are at distance r consists of two terms:
(i) the attractive van der Waals force, which decays with distance like 1/r^7, hence is immeasurable at the distance of 1m but noticeable as friction at close to contact distance. (It is attractive although the electrons carry the same negative charge since it also contains the effects of the positive charge of the nucleus.)
(ii) the repulsive (approximate hard core) force, which decays with distance like 1/r^11 (or so), hence is immeasurable already at distances just beyond contact but gets very strong at contact distance, and ensures that solid matter cannot penetrate other solid matter.
The same holds for fluid matter - liquids and gas, but there the molecules are so weakly held together that the matter simply gives way to the contact motion
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