The macroscopic elastic modulus and yield strength of solid-wall nickel gyroids and hollow-wall graphene gyroids of cell size 60 nm are deduced from indentation tests on a thin coating of the gyroids, with suitable interpretation by finite element simulations. The solid-wall nickel gyroids are fabricated by the self-assembly of a triblock copolymer, followed by the chemical vapour deposition of a graphene film onto this catalytic template. The nano-indentation response of the gyroid-based coatings was measured using a Berkovich indenter. In order to interpret the indentation response, two sets of finite element simulations were performed: periodic cell calculations in order to deduce the effective macroscopic properties in terms of the relative density and cell wall properties of the lattice, and then indentation simulations of a continuum with the effective properties of the gyroid. Despite the knockdown in modulus and strength of the graphene gyroid lattice due to waviness of the layered cell walls, the structure remains remarkably strong due to nanoscale size effects.