Zero dimensional points do not take up space, so then wouldn't everything in the universe be literally empty? Or is there something that I'm missing?
Although it's commonly said that fundamental particles are point particles you need to be clear what this means. To measure the size of the particle to within some experimental error $d$ requires the use of a probe with a wavelength of $\lambda=d$ or less i.e. with an energy of greater than around $hc/\lambda$. When we say particles are pointlike we mean that no matter how high the energy of your probe, or how small its wavelength, you will never measure a particle radius greater than your experimental limit $d$. That is the particle will always appear pointlike no matter how precise your experiment is.
But this does not mean that the particles are actually zero dimensional, infinite density, dots whizzing around. An elementary particle does not have a position in the way we think of a macroscopic object as having a position. It is always delocalised to some extent, i.e., it exists across a region of some non-zero volume. More precisely the probability of finding the particle is non-zero anywhere within that region.
So an atom is not empty space. The usual analogy is that it is a fuzzy blob, and actually that's a not a bad metaphor. If we take any small volume $\mathrm dV$ within the atom then the probability of finding the electron in that region is given by:
$$ P = \int \psi^*\psi\,\mathrm dV $$
where $\psi$ is the wavefunction describing the electron in the atom. And since this probability is just the charge density that means the charge density varies smoothly throughout the atom.
It is important to be clear that this is not just some time average due to the electron whizzing about the atom very fast. It is not the case that the electron has a precise position in the atom and our probability is some time average. The electron genuinely has no position in the usual macroscopic sense.
Because of the Pauli exclusion principle, it's extremely difficult to compress atomic matter beyond a certain density. It's not impossible, because there are always higher-energy electron states available, but there's a very strong force opposing it (called electron degeneracy pressure).
This is what it means for space to be full. If you define "empty space" in such a way that atoms are empty space, then atoms are empty space, but also the notion of "empty space" becomes useless because all space is empty. The idea of empty and occupied space long predates the modern understanding of fundamental particles. The job of science is to explain why the world is the way it is—for example, why you can't walk through walls—not to give new meanings to existing words.
Yes, elementary particles such as electrons and quarks (inside protons) are point-like or at least, their internal structure is incomparably smaller than the size of the atom. So the atom is mostly empty space.
However, that doesn't mean that atoms may penetrate each other. Matter is impenetrable because of a combination of
the uncertainty principle that says that the electrons can't be simultaneously sitting at/near the nucleus and have a small velocity (and kinetic energy), so the typical distance of an electron from a nucleus is finite (comparable to what is then interpreted as the size of the atom)
the Pauli exclusion principle that says that electrons can't occupy the same state. For this reason, even though the space in the atom is "empty", it is not possible for many electrons to occupy the same space.
For those reasons, atoms, although they're empty space, don't allow other atoms to overlap with them – and that's why atoms always "push" on other nearby atoms.
What we intuitively think of as "solid objects" are actually electromagnetic force-fields repelling each other. So you are correct; atoms are 'empty' in that they contain no solid objects or things. On the other hand, they are 'full' of basic force field which, in the aggregate, on a macro-scale, creates the illusion of 'solidity' that is what we perceive to be solid objects.
Another issue is that the forces between elementary particles in an atom introduce some characteristic length scales. For example, although quarks are possible point particles, the protons and neutrons in which they reside are about 2 femtometres wide because the forces between quarks prevent further compression. This gives nuclei a femtometre-scale size, but an atom is picometres wide due to the orbital radius of electrons. An atom therefore functions like a sphere with a certain chemical valency. In each case, the occupied space is filled with a forcefield of a certain mass-energy density.