According to our current understanding of physics, fundamental particles do not "touch" in the classical sense of two macroscopic objects coming into contact with each other. Instead, particles interact through fundamental forces, such as electromagnetic, weak, strong, and gravitational forces. These forces act over a distance and mediate the interactions between particles.
At the quantum level, particles are described by quantum field theory, where they are treated as excitations of underlying fields. These fields permeate all of space, and when particles interact, they exchange virtual particles, which are fluctuations in their respective fields. For example, in the case of electromagnetic interactions, particles exchange photons, which are quanta of the electromagnetic field.
Even when particles are very close to each other, there is a repulsive force called the Pauli exclusion principle that prevents identical particles, such as electrons, from occupying the same quantum state. This principle arises from the quantum nature of particles and ensures the stability of matter. Consequently, particles cannot occupy exactly the same position and momentum simultaneously.
In summary, at the quantum level, particles interact through fundamental forces and exchange virtual particles, but they do not "touch" in the way macroscopic objects do. Their interactions occur through fields and are mediated by forces acting over a distance.