First, point particles occupy space. The point particle at (1,1,1) in a 3D coordinate system occupies the space precisely at (1,1,1). Points are 0-dimensional, with a physical volume, surface area, and perimeter of 0. They, and any countable collection of them, have a measure of 0 under a standard measure but can have other measures under other choices.
Secondly, whether or not point particles exist in modern theories of physics is not entirely clear. It is true that, according to the wave/particle duality of quantum mechanics, Heisenberg’s Uncertainty Principle, and the fact that particles have a calculable de Brogle wavelength, particles seem to act as if they are not entirely point-like.
On the other hand, electrons, for example, have no internal structure and behave like point particles in other ways. Traditional interpretations of the collapse of the wave function conclude electrons are point particles after a measurement or observation. The details get hazy, dependent upon interpretation, and even philosophical very quickly. (Eg. For how long after? Decoherence anyone? How does this not violate the Uncertainty Principle and is the Uncertainty Principle epistemic or ontological?)
Nonetheless, modern theories of physics are so radically different from what anyone could have predicted before about 1870 (to be safe) that every philosopher who philosophized about physics was proven incorrect in countless foundational ways.
Update in response to other posted answers
“Dimensionless point particles do not exist in physics” is false.
After all, a particle, by its nature, is localized at a point, whereas the wave function (as it’s name suggests) is spread out in space (it’s a function of x, for any given time t). How can such an object represent the state of a particle? The answer is provided by Born’s statistical interpretation of the wave function, which says that [|psi| squared] gives the probability of finding a particle at point x at time t
Introduction to Quantum Mechanics, Second Edition. David J. Griffiths.
This is a canonical standard-bearer for quantum mechanics. Many other places in the text explicitly refer to a particle as being a point (in the collapse of the wave function).