Hyperbolic (or indefinite) materials that have been extensively studied in artificial electromagnetic metamaterials are attracting increasing interest due to their unusual optical properties, which enable useful functionalities in emerging devices based on light. Natural hyperbolic materials have many obvious advantages over metamaterials, most of which are inorganic. Here, using first-principles calculations, we demonstrated a metal-organic framework in the form of a natural hyperbolic material, Cu-benzenehexathial (Cu-BHT), which has been synthesized and proved to be a superconductor in recent experiments. Cu-BHT exhibits hyperbolic dispersion, i.e., all-angle negative refraction over a broad frequency from near-IR (∼1.4 μm) to the visible regime (∼750 nm). The operating hyperbolic window can be effectively tuned by applying uniaxial strain along the optical direction, constructing van der Waals heterostructures with BN or graphene or electron doping. This work is expected to offer a promising platform for the design of high-performance natural hyperbolic materials with great potential for applications.