Middle-aged divorcee Simon Wells (Macdonald Carey
) is on a boating holiday in the town of Weymouth on the south coast of England. There he meets Joan (Shirley Anne Field
), a 20-year-old girl, who lures him into a mugging staged by her biker brother King (Oliver Reed
) and his gang. Wells is beaten up and robbed.
Later Joan approaches Wells on his small boat. He is prepared to forgive and forget, she implies that he asked for it after trying to pick her up. At that moment they are threatened by King and his gang. Wells sets off on his boat and calls on Joan to join him which she does, defying her over-protective (and rather incestious) brother.
After drifting out to sea, Joan decides to go back and confront King, who has previously locked her into a cupboard when she has come close to other men. King and his gang keep a watch-out for them.
At night Wells and Joan return to a quiet part of the mainland and make love in a cliff top house surrounded by curious sculptures. Caught up and surrounded by the gang, the couple escape into a nearby military base where guards and dogs are soon in pursuit of them all.
The house was rented by sculptress Freya Neilson (Viveca Lindfors
) whose lover, Bernard (Alexander Knox
), is a scientist who runs the base. He will not discuss his work, warning her that he "might be condemning her to death".
Wells and Joan make their way down the cliff face. King goes after them. There they discover a network of caves and bunkers occupied by a small group of young boys and girls, aged about 11. Although well-dressed, cared-for and educated, it is clear that they do not know much about the outside world. The children's skin is also cold at the touch.
In fact the children are confined to the bunker which is the underground part of the military base. The premises, which double as a schoolroom and living quarters, are filled with surveillance cameras. Access to the children is limited though they are occasionally visited by men in radiation suits. Bernard gives them lessons via a TV monitor, but dismisses some of their more searching questions, saying that they will learn the answers "when the time comes".
The children do not even know much about where they are. One of them actually believes that they are in a spaceship and are to populate a distant planet.
The children have a small hideout in a connecting cave where they keep pictures and mementoes of people they think are their parents. They believe that it is a safe and secret place but Bernard and his associates are well aware of it. Bernard, who is, in his own way, genuinely fond of the children, has allowed them this degree of privacy in spite of objections by the head of security Major Holland (Walter Gotell
Time passes and Wells, Joan and King feel increasingly unwell, but Wells and Joan have promised to help the children escape and pressure King into helping them. The children keep them fed by ably smuggling food pass the surveillance cameras into the hideout.
Since security has been unable to find the intruders, Bernard tells the children via the TV monitor that he knows their secret and asks that they give up the grown-ups that they are hiding there. He reminds them of a rabbit they once acquired as a pet but which later died and warns them that the same might happen to their would-be rescuers. The children rebel and destroy the cameras.
Men in radiation suits then enter the premises but are overpowered and even killed by Wells and King. Using a Geiger counter, Wells discovers that it is in fact the children themselves who are radioactive. Nevertheless he agrees to break them out. They escape the bunker and face the world, the Sun and flowers for the first time.
But before they can take it all in, they are rounded up and, kicking and screaming, are taken back to the bunker by other men in radiation suits. Freya Neilson, the sculptress, witnesses the events.
King drives off in a sports car accompanied by Henry (Kit Williams
), one of the children. Weakened by the radioactivity, King orders Henry out of the car saying that he is "poison". Henry is then grabbed by pursuers and forcibly taken back to the bunker in a helicopter.
Chased by another helicopter, King crashes his car into a river. Joan and Wells are allowed to leave on his boat, but the sickness catches up with them and the boat goes round and round in the sea monitored by a military helicopter, which will destroy the boat once its occupants are dead.
Bernard's only real regret about the whole affair is that "his" children now know for sure that they are "freaks" and prisoners and will be more difficult to control. He explains to Neilson that they were radioactive from birth. The plan is to release them "when the time comes", i.e. the inevitable nuclear war. Their bodies will resist the nuclear fallout and the human race will continue. Now that Neilson knows his secret, Bernard kills her. (In the 1950s and early 60s, the threat of nuclear war was very real, a matter of "when" not "if", and culminated in the Cuban Missile Crisis.)
The film ends with the children desperately and vainly calling for help from their cliff side prison.