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Harry Turner and Chester Babcock are two vaudeville song-and-dance men touring the Far East.  While hawking a phony do-it-yourself space kit in India, Chester is knocked unconscious and loses his memory.  Harry takes him to a Tibetan Lamasery, where he is given a special herb that cures his amnesia and provides him with a photographic memory.

At the airport, Chester meets Diane, a beautiful spy for the Third Echelon, an organization of mad scientists planning to conquer the universe.  She mistakes him for a photographer assigned to copy a secret Russian formula for rocket fuel, and Chester is able to memorize the information merely by glancing at the equations.

Later, the two men are captured by the organization and inadvertently become substitutes for a pair of apes on a moon reconnaissance flight when they try to escape.

Upon their return, the Third Echelon decides to sacrifice them to science but quarters the two men in a harem to make their final hours more pleasant.  They evade their captors and flee through Hong Kong with Diane, who has come to realize that the organization's leader is a madman.  An old friend, Dorothy Lamour, who is appearing at a local nightclub, helps the trio outwit their pursuers and alert the Hong Kong police.  As the Third Echelon headquarters are raided, however, Harry, Chester, and Diane are trapped in a rocket that takes them to Plutonlum, a remote planet, where they are met by Frank Sinatra and Dean Martin, who have arrived in another spacecraft.