A storage system for storing a cryogenic medium. The storage system includes a storage container operable to receive the cryogenic medium; a first removal line forming a fluid-conducting connection from an interior of the storage container to a first consumer connection for connecting a consumer device that uses the cryogenic medium; a first controllable line shut-off valve arranged in the first removal line; a first heat exchanger arranged in the first removal line; a second removal line, redundant to the first removal line, forming a second fluid-conducting connection from the interior of the storage container to a second consumer connection for connecting the consumer device; a second controllable line shut-off valve, redundant to the first controllable line shut-off valve, arranged in the second removal line; and a second heat exchanger, redundant to the first heat exchanger, arranged in the second removal line.