We consider the capacity region of a $K$-user multiple access channel (MAC) with energy harvesting transmitters. Each user stores and schedules the randomly arriving energy using an energy buffer. Users can also perform energy cooperation by transmitting energy to other users or receiving energy from them. We derive the capacity region of this channel and show that 1) the capacity region coincides with that of a traditional $K$-user Gaussian MAC with energy cooperation, where the average power constraints are equal to the battery recharging rates of the energy harvesting case; 2) each rate on the capacity region boundary can be achieved using the save-and-forward power control and a fixed energy cooperation policy.