The Muon Ionization Cooling Experiment (MICE) has been built at the STFC Rutherford Appleton Laboratory to demonstrate the principle of muon beam phase-space reduction via ionization cooling. Muon beam cooling will be required at a future proton-derived neutrino factory or muon collider. Ionization cooling is achieved by passing the beam through an energy-absorbing material, such as liquid hydrogen, and then re-accelerating the beam using RF cavities. This paper describes the hydrogen system constructed for MICE including: the liquid-hydrogen absorber, its associated cryogenic and gas systems, the control and monitoring system, and the necessary safety engineering. The performance of the system in cool-down, liquefaction, and stable operation is also presented.