A bearing system is provided that includes a stationary structure, a rotating structure, a magnetic-foil bearing and a cooling jacket. The rotating structure is rotatable about an axis. The magnetic-foil bearing radially supports the rotating structure within the stationary structure. The magnetic-foil bearing includes a magnetic bearing stator, a magnetic bearing rotor and a foil bearing. The foil bearing is disposed radially between the magnetic bearing stator and the magnetic bearing rotor. The cooling jacket circumscribes and radially engages the magnetic bearing stator. A cooling circuit is configured to flow cooling fluid within the cooling jacket to cool the magnetic bearing stator.