The realization of levels of stretchability that extend beyond intrinsic limits of bulk materials is of great importance to stretchable electronics. Here we report large-area, three-dimensional nano-architectures that achieve this outcome in materials that offer both insulating and conductive properties. For the elastomer poly(dimethylsiloxane), such geometries enhance the stretchability and fracture strain by ~62% and ~225% over the bulk, unstructured case. The underlying physics involves local rotations of narrow structural elements in the three-dimensional network, as identified by mechanical modelling. To demonstrate the applications of three-dimensional poly(dimethylsiloxane), we create a stretchable conductor obtained by filling the interstitial regions with liquid metal. This stretchable composite shows extremely high electrical conductivity (~24,100 S cm⁻¹) even at strains >200%, with good cyclic properties and with current-carrying capacities that are sufficient for interconnects in light-emitting diode systems. Collectively, these concepts provide new design opportunities for stretchable electronics.