There is strong recent interest in ultrathin, flexible, safe energy storage devices to meet the various design and power needs of modern gadgets. To build such fully flexible and robust electrochemical devices, multiple components with specific electrochemical and interfacial properties need to be integrated into single units. Here we show that these basic components, the electrode, separator, and electrolyte, can all be integrated into single contiguous nanocomposite units that can serve as building blocks for a variety of thin mechanically flexible energy storage devices. Nanoporous cellulose paper embedded with aligned carbon nanotube electrode and electrolyte constitutes the basic unit. The units are used to build various flexible supercapacitor, battery, hybrid, and dual-storage battery-in-supercapacitor devices. The thin freestanding nanocomposite paper devices offer complete mechanical flexibility during operation. The supercapacitors operate with electrolytes including aqueous solvents, room temperature ionic liquids, and bioelectrolytes and over record temperature ranges. These easy-to-assemble integrated nanocomposite energy-storage systems could provide unprecedented design ingenuity for a variety of devices operating over a wide range of temperature and environmental conditions.