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, civilisation 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.
Curiously, when the TARDIS rematerialises, they are still on the Ark. However, seven hundred years have passed and there has been a major change: the Monoids are in control. They have completed the statue in the image of themselves, having staged a coup during the long journey. This was made possible by a genetic weakness introduced into the humans, but not the Monoids, by a second wave of the cold virus 700 years earlier. The Monoids also now have voice communicators and use numerical emblems to distinguish each other. The humans are now little more than slaves, with the odd exception like the collaborator subject Guardian Maharis, and have little hope of change. The Doctor and his friends encounter the Monoid leadership, installed in a throne room on the Ark, after which they are sent to the security kitchen to help prepare meals for the Monoids. Two humans, Manissa and Dassuk, believe the moment of their liberation is at hand. Steven tries to help them in a revolt which is unsuccessful.
The arrival on Refusis is close at hand and a landing pod is prepared. Monoid 1 wants to make sure that the new world is inhabited only by Monoids, despite promises that the human population will be allowed to live there too. A landing party is assembled – the Doctor, Dodo, Monoid 2 and a subject Guardian named Yendom – and they soon reach Refusis II and start to investigate. A stately castle which seems to be unoccupied is in fact the home to the invisible Refusians, giant beings rendered invisible by solar flares. They welcome their guests and have been expecting them but only want to share the planet with other peaceful beings. Monoid 2 and Yendom flee the castle, and en route Yendom realises the humans will not be allowed to reach Refusis with the Monoids. Monoid 2 kills him and is shortly afterward killed himself when the landing pod explodes.
The tension of the situation foments dissent in the Monoid ranks, with Monoid 4 openly opposing Monoid 1's plans to abandon the humans and colonise Refusis without more checks on the planet. Three launchers are sent to the planet, Monoids 1 and 4 commanding them, and when the crews emerge Monoid 4 interprets the destroyed landing pod as evidence of the danger that Monoid 1 has led them to. A civil war erupts between the two Monoid factions. The Doctor, Dodo and a Refusian use the confusion to steal one of the launchers and pilot back to the Ark.
The Monoids have placed a bomb on board the ship and plan to evacuate soon to the planet surface, leaving the humans to die on the spaceship. Word of this threat spreads and spurs a human rebellion. The arrival of the Doctor and the Refusian spur things along, and they soon realise the bomb has been placed in the head of the statue. Thankfully the Refusian is able to help dispose of the statue into space before the bomb explodes. The humans now begin to land on Refusis themselves, having been offered support on peaceful terms by the Refusians. Many of the Monoids have been killed in their civil war and those that remain are offered peaceful settlement alongside the other two species.
Once more the TARDIS departs, and this time the curiosity is that the Doctor simply vanishes from the TARDIS control room…