The short answer
The People of the Cave are young believers in Surah 18 who fled a persecuting king and slept in a cave for "three hundred years and add nine." Christians tell the same story as the Seven Sleepers of Ephesus.
Surah 18 is named for them. A group of young men flee a persecuting king and take refuge in a cave. God puts them to sleep, turning them from side to side, while their dog lies stretched at the entrance. When they wake, they think they have slept a day or part of a day. One of them goes to the city with an old silver coin to buy food, and their secret is discovered.
The Quran is strikingly reluctant to fix details. It reports that people argue about their number, three, five, or seven, with the dog making one more, and says only God knows. On the duration it says: "And (it is said) they tarried in their Cave three hundred years and add nine," then adds that God knows best how long they stayed. Three hundred solar years come to almost exactly 309 lunar years, which is the usual explanation for the extra nine.
The Christian version, the Seven Sleepers of Ephesus, sets the story under the emperor Decius around 250 CE, when seven young Christians are walled up in a cave. They wake in the reign of Theodosius II, in the 5th century, when a landowner opens the cave, and their existence proves the resurrection of the body. The Syriac poet Jacob of Serugh wrote an early version, and Gregory of Tours spread it in the West.
The tale is one of the most widely shared legends between Christianity and Islam, and several caves in Turkey, Jordan, and elsewhere claim to be the site. For readers of time-slip and suspended-animation stories, it is an early account of people held outside of time while the world moved on.