In nuclear physics, volume is usually defined in terms of the mean square radius of a particle's position wavefunction, as my understanding goes. It's a rough measure of the region of space in which one would most often expect the particle to be detected. As far as describing a physical volume occupied by a point particle, I doubt any physically meaningful definition could be made. In the case of bosonic particles, you can have an unlimited number of them occupying the same space and having the same quantum numbers, so the idea of a particle occupying a region of space all to itself doesn't hold up. As I understand it, particles mathematically interact as if they only exist at a single point, but these interactions are then summed over every point in the universe, because the particle has a probability amplitude for existing and interacting at each of these points.