We introduce composite materials made by reversibly assembling a three-dimensional lattice of mass-produced carbon fiber-reinforced polymer composite parts with integrated mechanical interlocking connections. The resulting cellular composite materials can respond as an elastic solid with an extremely large measured modulus for an ultralight material (12.3 megapascals at a density of 7.2 milligrams per cubic centimeter). These materials offer a hierarchical decomposition in modeling, with bulk properties that can be predicted from component measurements and deformation modes that can be determined by the placement of part types. Because site locations are locally constrained, structures can be produced in a relative assembly process that merges desirable features of fiber composites, cellular materials, and additive manufacturing.