Copper-based high-strength nanofilamentary wires reinforced by tantalum nanofilaments were prepared by severe plastic deformation (repeated hot extrusion and cold drawing steps) to be used in the windings of high-pulsed magnets. This application requires a complete characterization of the microstructure and the strength and their relationship for further optimization: after heavy strain, the Cu matrix is nanostructured and the Ta nanofilaments develop a strong ribbon-like shape resulting in an early microstructural refinement. The macroscopic strength is greater than rule-of-mixture predictions as confirmed by nanohardness values. The observed size effect is related to the dislocation starvation in the nanostructured materials combined with the barrier role of Cu/Ta interfaces. The strengthening is lower, however, as expected, because of the distorted ribbon morphology of the Ta filaments preventing them from behaving as nanowhiskers, as Nb fibers do in Cu/Nb wires. This shows that size and geometry play key roles in the plasticity of nanomaterials.