Emerging demands for national security, transportation, distributed power, and portable systems call for energy storage and conversion technologies that can simultaneously deliver large power and energy densities. To this end, here we report three-dimensional Ni/Ni(OH)2 composite electrodes derived from a new class of multi-phase soft materials with uniform, co-continuous, and tunable internal microdomains. These remarkable morphological attributes combined with our facile chemical processing techniques allow the electrode's salient morphological parameters to be independently tuned for rapid ion transport and a large volumetric energy storage capacity. Through microstructural design and optimization, our composite electrodes can simultaneously deliver energy densities equal to that of batteries and power densities equivalent to or greater than that of the best supercapacitors, bridging the gap between these modern technologies. Our synthesis procedure is robust and can be extended to a myriad of other chemistries for next generation energy storage materials.