An aircraft fuel cell system includes a fuel cell which has an oxidant inlet for supplying an oxygen-containing medium to the fuel cell. An oxidant supply line has a first end which is connected to the oxidant inlet of the fuel cell. A second end of the oxidant supply line is connectable to a used air outlet of a cabin of the aircraft. A heat exchanger is located in the oxidant supply line and thermally couples the oxygen-containing medium flowing through the oxidant supply line and a second medium flowing through an air conditioning process air line of an air conditioning unit, the heat exchanger being located downstream of a cabin air compressor in the air conditioning process air line.