the degenerate Fermi gas
Cool a gas of fermions — electrons, say — toward absolute zero. Because the Pauli exclusion principle forbids two of them from sharing a state, they cannot all sink into the ground state the way a classical gas would. Instead they stack up, one per state, filling every level from the bottom to a sharp cutoff. Even at T = 0 the gas seethes with enormous kinetic energy. This is the degenerate Fermi gas, and it is the model behind conduction electrons in metals and the matter inside white dwarfs.
'Degenerate' here means quantum-degenerate: the temperature is so low, kT << E_F, that the Fermi-Dirac distribution is essentially a step function and quantum statistics dominate completely. At T = 0 all states are filled up to the Fermi energy E_F and empty above it. Filling momentum space out to the Fermi momentum p_F gives, for a spin-1/2 gas of density n, a Fermi energy E_F = (hbar^2/2m)(3 pi^2 n)^(2/3). The states form a filled sphere in momentum space, the Fermi sea, bounded by the Fermi surface. At small but finite T only electrons within about kT of E_F can be excited, which the Sommerfeld expansion turns into concrete predictions: an electronic heat capacity linear in temperature, C proportional to T, and a temperature-independent (Pauli) paramagnetic susceptibility.
The degenerate Fermi gas explains why metals have a small linear heat capacity rather than the classical (3/2)Nk, why they conduct, and how white dwarfs and neutron stars resist gravitational collapse through degeneracy pressure. An essential clarification: 'degenerate' has nothing to do with degenerate energy levels in atomic physics — it refers to the gas being quantum-degenerate. And the model is an idealization of NON-interacting fermions; Coulomb repulsion between real electrons is handled by the more sophisticated Landau Fermi-liquid theory, which remarkably preserves the qualitative picture.
The conduction electrons in sodium have a density giving a Fermi energy of about 3.2 eV, equivalent to a Fermi temperature T_F = E_F/k of roughly 37000 K. Room temperature (300 K) is thus deep in the degenerate regime, kT/E_F is about 0.008, which is exactly why the electron gas contributes so little to the specific heat.
At room temperature a metal's electrons are a deeply degenerate quantum gas, not a classical one.
The huge zero-point kinetic energy is a purely quantum, purely statistical effect of the exclusion principle, not thermal motion — a degenerate gas at T = 0 still has enormous pressure. Do not confuse 'degenerate gas' (quantum-degenerate) with 'degenerate levels' (equal-energy states) in spectroscopy.