On a near-duplicate of Earth, a failed life-prolongation experiment had killed every adult centuries earlier, leaving long-lived children who died of the disease at puberty.
Answering an old distress call, the Enterprise found a planet almost identical to Earth whose adults had all died about three centuries earlier. A virus created in a life-prolongation project had slowed the aging of children enormously, but it killed them horribly once they reached adolescence. The surviving children, who called themselves the Onlies, lived wild among the ruins and feared all grown-ups.
The landing party, including Kirk, Spock, McCoy and Rand, caught the disease. With the help of Miri, a girl on the edge of adulthood, Kirk won the children's trust and recovered the laboratory equipment McCoy needed. McCoy tested the vaccine on himself, it worked, and the Federation sent help.