Can there exist a particle/object in the universe having mass but no volume? Is it possible that mass can exist without volume and density? We think we know that matter is anything having mass and that it occupies space, but is it possible that this statement is wrong?
In the standard model of particle physics which fits the data up to now elementary particles entering the lagrangian are point particles with mass.
The electron, for example is one of the elementary particles, and it does have a mass and the fit gives it 0 volume.
There are experiments which try to set limits to how small the volume of the electron is. The fact that the standard model fits a large number of measurements in elementary particles with zero point dimensions for the elementary particles can be considered as a measurement.
One should keep in mind though that the electron is a quantum mechanical entity , and follows quantum mechanical equations. If one tries to determine the size the probe that checks the size follows the Heisenberg uncertainty principle, which means to get very small limits one needs very high momentum probes.
String theories posit that elementary particles are vibrations on one dimensional strings, with a length smaller than the Planck length. Still, a line has no volume, even if this is so.
Your question states that
We think we know that matter is anything having mass and that it occupies space
but in fact, we know better than that. We have good reason to believe that fundamental particles are point-like. In other words, they have no internal structure, size, or volume. And they indeed have mass. We have a theoretical understanding (in local Quantum Field Theory) and experimental evidence (from collider experiments) for objects with mass but no volume.
This isn't the final word, though, because it's quite possible (and some might say very likely) that particles that appear point-like in our experiments have substructure, including a characteristic size, that would be revealed in very high-energy experiments. I believe that to be the case in string theory.
In summary then, we have no evidence that particles are not point-like and a solid theoretical understanding of point-like particles, but there is motivation for considering internal structure, which could be revealed in very high-energy experiments.
I'll address your question a little different, because talking about volumn and particles is problematic in many ways.
Let's phrase your question "can there be two particles with mass be at the same place". The answer is yes. There are two types of particles:fermions and bosons. While fermions (electrons, protons) repel each other (not only because of the charge!) Bosons can be packed as dense as you want. For example photons are bosons, but even fermions can be coupled to form bosons, like the helium atom. You see, this is a little different from your question since helium as clearly a volumn but photons don't.
Based on the latest breakthroughs in particle physics, the answer is a plain NO - it's not possible for a massive particle to have no volume. In fact, it is NOT possible for any particle, whether massive or massless, to have zero volume. ALL particles have a certain volume, no matter how small beyond observation.
On the contrary, mass is an intrinsic property of a particle that arise from the Higgs mechanism. Particles that do not interact are massless, those that do exhibits mass.
However, as research continue to grow and our knowledge base is updated, the answer can change, or be uncertain, much like the uncertainty principle. =)
Is it possible for an object to have mass but zero volume?
Can there exist a particle/object in the universe having mass but no volume?
Is it possible that mass can exist without volume and density?
We think we know that matter is anything having mass and that it occupies space, but is it possible that this statement is wrong?
Let's start with an E=hc/λ photon. It has a non-zero energy, and a non-zero wavelength. So it's a wave, not a point-particle. Now have a look at wind waves on Wikipedia. See the gif with the red dots, and crop it to remove the surface and emulate a wave in space:
GNUFDL image by Kraaiennest, see Wikipedia Commons
Now think about pair production where we create matter from light. Then remember electron diffraction and the wave nature of matter. And most important of all, remember that it's quantum field theory, not quantum point-particle theory. The electron is not some speck that has a field. Instead field is what it is. And this field doesn't have an edge or a surface. It gets weaker away from the centre, but it doesn't stop, ever. The electron just isn't a point particle, despite what some people say. Nor is a proton, and nor is a hydrogen atom. But a hydrogen atom is just an electron and a proton, and they are "just their fields". So the hydrogen atoms's gravitational field isn't something distinct from the hydrogen atom. It's part of what it is. The same applies to a star, and to a black hole.
So there are no zero-volume masses.