Like an ideal gas, a Fermi gas is composed of non-interacting particles. This is typically an idealization--few gases are composed of entirely non-interacting particles--but it's no more of an idealization than the classical ideal gas is.
The differences between Maxwell-Boltzmann statistics (which yields the classical ideal gas), Bose-Einstein statistics (which yields the Bose gas) and Fermi-Dirac statistics (the Fermi gas) are most easily seen at low temperatures and densities. At high temperatures and densities, both BE and FD statistics look like MB statistics--these conditions suppress most of the new, novel quantum effects.
At low temperatures, a classical ideal gas will have many states in which particles are spread out among energies and temperatures. This is a consequence of particles being distinguishable from one another. On the other hand, it does contain states in which all the particles are in the lowest energy (ground) state.
In BE statistics, the particles are all indistinguishable, and as a result, extreme distributions (e.g. all particles in the ground state) become more likely. With a slight energy difference between the ground state and the first excited state, it can happen that a huge proportion of particles stay in the ground state. This is Bose-Einstein condensation, and it is a characteristic feature of a Bose gas compared to the classical ideal gas.
In FD statistics, no two particles can occupy the same state. This imposes a serious constraint on the allocation of particles with energies, so that even at zero temperature, many excited states are occupied. This is called degeneracy pressure, and it is the characteristic feature of a Fermi gas.
All three of these gases--the classical ideal gas, the Bose gas, and the Fermi gas--are composed of particles that move around as you'd expect. The differences appear mostly at low temperatures and densities and are all consequences of how the particles can sit in various energy states and whether they are distinguishable.