Energy storage devices that provide high specific power without compromising on specific energy are highly desirable for many electric-powered applications. Here, we demonstrate that polymer organic radical gel materials support fast bulk-redox charge storage, commensurate to surface double layer ion exchange at carbon electrodes. When integrated with a carbon-based electrical double layer capacitor, nearly ideal electrode properties such as high electrical and ionic conductivity, fast bulk redox and surface charge storage as well as excellent cycling stability are attained. Such hybrid carbon redox-polymer-gel electrodes support unprecedented discharge rate of 1,000C with 50% of the nominal capacity delivered in less than 2 seconds. Devices made with such electrodes hold the potential for battery-scale energy storage while attaining supercapacitor-like power performances.