Robots made from reconfigurable modular units feature versatility, cost efficiency, and improved sustainability compared with fixed designs. Reconfigurable modules driven by soft actuators provide adaptable actuation, safe interaction, and wide design freedom, but existing soft modules would benefit from high-speed and high-strain actuation, as well as driving methods well-suited to untethered operation. Here, we introduce a class of electrically actuated robotic modules that provide high-speed (a peak contractile strain rate of 4618\% per second, 15.8-hertz bandwidth, and a peak specific power of 122 watts per kilogram), high-strain (49\% contraction) actuation and that use magnets for reversible mechanical and electrical connections between neighboring modules, thereby serving as building blocks for rapidly reconfigurable and highly agile robotic systems. The actuation performance of each hexagonal electrohydraulic (HEXEL) module is enabled by a synergistic combination of soft and rigid components; a hexagonal exoskeleton of rigid plates amplifies the motion produced by soft electrohydraulic actuators and provides a mechanical structure and connection platform for reconfigurable robots composed of many modules. We characterize the actuation performance of individual HEXEL modules, present a model that captures their quasi-static force-stroke behavior, and demonstrate both a high-jumping and a fast pipe-crawling robot. Using embedded magnetic connections, we arranged multiple modules into reconfigurable robots with diverse functionality, including a high-stroke muscle, a multimodal active array, a table-top active platform, and a fast-rolling robot. We further leveraged the magnetic connections for hosting untethered, snap-on driving electronics, together highlighting the promise of HEXEL modules for creating rapidly reconfigurable high-speed robots.