Modern robots lack the multifunctional interconnected systems found in living organisms and are consequently unable to reproduce their efficiency and autonomy. Energy-storage systems are among the most crucial limitations to robot autonomy, but their size, weight, material and design constraints can be re-examined in the context of multifunctional, bio-inspired applications. Here we present a synthetic energy-dense circulatory system embedded in an untethered, aquatic soft robot. Modelled after redox flow batteries, this synthetic vascular system combines the functions of hydraulic force transmission, actuation and energy storage into a single integrated design that geometrically increases the energy density of the robot to enable operation for long durations (up to 36 hours). The fabrication techniques and flexible materials used in its construction enable the vascular system to be created with complex form factors that continuously deform with the robot's movement. This use of electrochemical energy storage in hydraulic fluids could facilitate increased energy density, autonomy, efficiency and multifunctionality in future robot designs.