  We study metamaterials known as hyperbolic media that in the usual
local-response approximation exhibit hyperbolic dispersion and an associated
broadband singularity in the density of states. Instead, from the more
microscopic hydrodynamic Drude theory we derive qualitatively different optical
properties of these metamaterials, due to the free-electron nonlocal optical
response of their metal constituents. We demonstrate that nonlocal response
gives rise to a large-wavevector cutoff in the dispersion that is inversely
proportional to the Fermi velocity of the electron gas, but also for small
wavevectors we find differences for the hyperbolic dispersion. Moreover, the
size of the unit cell influences effective parameters of the metamaterial even
in the deep sub-wavelength regime. Finally, instead of the broadband
super-singularity in the local density of states, we predict a large but finite
maximal enhancement proportional to the inverse cube of the Fermi velocity.
