Excitations of free electrons and optical phonons are known to permit access to the negative real part of relative permittivities (ϵ^{'}<0) that yield strong light-matter interactions. However, negative ϵ^{'} arising from excitons has been much less explored. Via development of a dielectric-coating based technique described herein, we report fundamental optical properties of two-dimensional hybrid perovskites (2DHPs), composed of alternating layers of inorganic and organic sublattices. Low members of 2DHPs (N=1 and N=2) exhibit negative ϵ^{'} stemming from the large exciton binding energy and sizable oscillator strength. Furthermore, hyperbolic dispersion (i.e., ϵ^{'} changes sign with directions) occurs in the visible range, which has been previously achieved only with artificial metamaterials. Such naturally occurring, exotic dispersion stems from the extremely anisotropic excitonic behaviors of 2DHPs, and can intrinsically support a large photonic density of states. We suggest that several other van der Waals solids may exhibit similar behaviors arising from excitonic response.