Almost ten million years in the future, the TARDIS materialises on a vast spacecraft including its own miniature zoo and arboretum. The First Doctor and Steven Taylor are still explaining the basics of their time travel ability to new companion Dodo Chaplet when she starts to show signs of a cold. It is only a matter of time before they are found and taken to the control chamber of the vessel. Their captors are the mute Monoids, seemingly identical alien beings with a single eye. The Monoids live in peace alongside the humans who command the spaceship, their own planet having been destroyed, but often do much of the menial work. The humans in charge of the ship explain that the Earth is about to be destroyed because of the expansion of the sun, and that this ship is an Ark sent into space with the last remnants of humanity, civilization and various forms of flora and fauna. The human Guardians in charge of the craft run a tight ship: failure to conform to rules means either death or miniaturisation until they reach their destination, an Earth-like planet called Refusis II, which takes nearly 700 years to get to. As an amusement during the journey a vast statue is being carved by hand, depicting a human being.
Dodo's cold has now spread amongst the Monoid and human populations, but regrettably, they have little natural immunity. When the Commander of the Ark collapses with the malady, the whole ship is placed on alert as Zentos, the Deputy Commander is suspicious of the travellers and believes they have deliberately infected the ship. When the first Monoid dies, there is little the Doctor can say to pacify the angry Guardians. Zentos places the Doctor, Steven and Dodo on trial for their crimes, with a young Guardian called Manyak and the Commander's daughter Mellium as defence. Steven acts as the first defence witness, attacking the closed nature of the minds of the Guardians, but exhausts himself in the process and collapses with the fever. His words have no impact on Zentos, who orders their execution, but the ailing Commander intervenes to protect the three travellers and permit them access to medical equipment to devise a cure to the cold. The Doctor is thus able to recreate the cold vaccine from the membranes of animals on the craft, and this is administered throughout the crew. The Commander, Steven and the others infected are soon on the road to recovery. Their work done, the trio have only time to observe the end of Earth on the long-range scanner before the Doctor leads them back to the TARDIS.