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 subwavelength regime. Finally, instead of the broadband supersingularity in the local density of states, we predict a large but finite maximal enhancement proportional to the inverse cube of the Fermi velocity.