We study the electronic band structure and optical conductivity of an array of tunnel-coupled array of graphene nanoribbons. We show that due to the coupling of electronic edge states for the zigzag nanoribbon structure, the Fermi surface can become a hyperbola similarly to the case of the layered metal-dielectric structures, where the hyperbolic isofrequency contours originate from the coupling of localized surface plasmon polaritons. Moreover, we show that for both types of the ribbon edge, the optical response of the structure can be characterized by a uniaxial conductivity tensor, having principal components of the different signs. Therefore, the tunnel-coupled nanoribbon array can be regarded as a tunable hyperbolic metasurface.