We study the fermionic King model which may provide a relevant model of dark matter halos. The exclusion constraint can be due to quantum mechanics (for fermions such as massive neutrinos) or to Lynden-Bell's statistics (for collisionless systems undergoing violent relaxation). This model has a finite mass. Furthermore, a statistical equilibrium state exists for all accessible values of energy. Dwarf and intermediate size halos are degenerate quantum objects stabilized against gravitational collapse by the Pauli exclusion principle. Large halos at sufficiently high energies are in a gaseous phase where quantum effects are negligible. They are stabilized by thermal motion. Below a critical energy they undergo gravitational collapse (gravothermal catastrophe). This may lead to the formation of a central black hole that does not affect the structure of the halo. This may also lead to the formation of a compact degenerate object surrounded by a hot massive atmosphere extending at large distances. We argue that large dark matter halos should not contain a degenerate nucleus (fermion ball) because these nucleus-halo structures are thermodynamically unstable. We compare the rotation curves of the classical King model to observations of large dark matter halos (Burkert profile). Because of collisions and evaporation, the central density increases while the slope of the halo density profile decreases until an instability takes place. We find that the observations are compatible with a King profile at, or close to, the point of marginal stability in the microcanonical ensemble. At that point, the King profile can be fitted by the modified Hubble profile that has a flat core and a halo in which the density decreases as $r^{-3}$. This is qualitatively similar to the Burkert profile.