Electric capacitors are commonly used in electronic circuits for the short-term storage of small amounts of energy. It is desirable however to use capacitors to store much larger energy amounts to replace rechargeable batteries. Unfortunately existing capacitors cannot store sufficient energy to be able to replace common electrochemical energy storage systems. Here we examine the energy storage capabilities of graphene nanocapacitors, which are tri-layer devices involving an Al film, Al2O3 dielectric layer, and a single layer of carbon atoms, i.e., graphene. This is a purely electronic capacitor and therefore it can function in a wide temperature interval. The capacitor shows a high dielectric breakdown electric field strength, of the order of 1000 kV mm-1 (i.e., 1 GV m-1), which is much larger than the table value of the Al2O3 dielectric strength. The corresponding energy density is 10-100 times larger than the energy density of a common electrolytic capacitor. Moreover, we discover that the amount of charge stored in the dielectric layer can be equal or can even exceed the amount of charge stored on the capacitor plates. The dielectric discharge current follows a power-law time dependence. We suggest a model to explain this behavior.