The King of the Potato People was a figment of Rimmer's fevered imagination.
When he had contracted the Holovirus, Rimmer descended into madness. He appeared to his shipmates in quarantine and told them that the King of the Potato People was demanding that they remain incarcerated for ten years. When Cat suggested that they could go to see the King, Rimmer asked if they had a magic carpet. Lister replied that they had a three-seater, which Rimmer then used as evidence that they had gone insane.
He sentenced them to go for two hours without oxygen. In light of this punishment, Cat still thought that their best bet was to plead with the King of the Potato People. ("Quarantine")