Captain James Webber, Peter Kirby, and Kurt Meyers, are lost in space and finally come across an asteroid 665 million miles from Earth as their fuel run low. They land the ship and confirm that the asteroid's gravity, atmosphere, and temperature are all like Earth's. When Kirby, the youngest crew member, prepares to go outside, Webber advises against it. However, Kirby points out that they have no choice. He opens the hatch and the crew is surprised to discover that they have landed at a farmyard.
As they explore, the three men spot a dog. However, it doesn't respond when they call it over. Examining the farm machinery, the science officer Meyers identifies it as dating back to the 20th century, 200 years before their current time. Kirby wonders if they have somehow traveled back in time, but Meyers points out that there are twin suns in the sky, meaning they are outside the solar system. Webber finds the farm owner, a man meaning on a pitchfork, but he doesn't respond to them. They examine the farmer but he is seemingly paralyzed.
The crew goes down the road from the farm and come to a bridge. A man is fishing by the side of the river, but when Kirby touches him, the fisherman falls over, unresponsive. They hear music playing and run into a nearby town where they find a frozen marching band. The music they are "playing" is piped in from loudspeakers, and a banner indicates that they are welcoming Mayor Finch. Going into the house, the crew finds the frozen mayor looking out from a balcony onto a paralyzed crowd of well-wishers. Meyers wonders if someone is trying to convince them that they're back on Earth using mental illusions, but Webbers notes that if that were the case, they would see images from their own time, not 200 years in the past. Meyers suggests that their personal time may have accelerated in some manner, and Kirby turns to a clock for confirmation. Much to their surprise, the hands are missing.
Webber orders the men to split up and search for any clues. He goes to a bar where a poker player is winning big and holds four aces and a joker in his hand. Meyers checks out a hotel room and finds an older man and a younger woman frozen in mid-dance as violinists serenade them. Kirby finds an exhibition hall and a beauty pageant where a homely woman receives the award over traditionally beautiful competitors. Kirby congratulates her but then loses and screams at the audience to say or do something. As he storms off, Kirby is unaware that one of the audience members is conscious and mobile, and examining the crewman with interest.
The three men meet after an hour on a quiet suburban street. As they approach a house, they notice an elderly man sitting on the porch, reading a paper. Much to their surprise, he lowers his paper and welcomes them. The man introduces himself as Jeremy Wickwire and tells them that they have nothing to be afraid of. He invites them inside and explains that the house was built for a Mr. Jacobson, but Jacobson changed his mind and is now in the medieval section. When the crew asks what he means, Wickwire explains that there are numerous sections covering all major historical periods. He informs them that the 20th century section they are now occupying is the most popular because it represents an idyllic period in Earth history.
Webber explains how their ship was damaged in a meteor shower and they were forced to land on the asteroid after six months in space because they have no way back. Wickwire asks if Earth eventually had a nuclear war, and Meyers tells him that it began in 1985. The resulting exchange destroyed much of the Earth's surface and they are just now starting to rebuild. As Wickwire offers to get them something to eat, an angry Kirby demands answers and Wickwire casually tells them that they're in a cemetery.
After getting food and drink, Wickwire serves the men and offers a toast to eternal peace. They join in the toast and then ask him for help. Before he answers their questions, Wickwire first asks them where they would be rather than anywhere else. Kirby says that they want to be on their spaceship going home and the other two men agree. Wickwire then asks them what year it was when they left Earth. Once he learns they departed in 2185, Wickwire explains that he thought they were people from Happy Glades, the world's greatest mortuary. The mortuary management came up with a plan where they would create tableaus for wealthy clients, granting each one their greatest wish in the afterlife. The client would then be embalmed and placed in the tableau, with wax dummies as the admirers as necessary. When Webber asks why they chose an asteroid instead of a desert on Earth, Wickwire points out that there was no guarantee of eternal peace on Earth, and that's what Happy Glades promised.
The crew wonder who Wickwire is, and the old man explains that he has been the caretaker since 1973. When Webber notes that would make the man over 200 years old, Wickwire sheepishly explains that in fact he is not a man but a facsimile that "sleeps" when not needed. When they are gone, he will go back to asleep again. The three men start to collapse and Wickwire explains that they won't be leaving, but they will be "gone" since they will die. The crew realizes that their drinks were poisoning and ask for the antidote. Wickwire tells them that there is no antidote, and that the eternifying fluid will preserve them forever. When the dying Meyers asks him why, Wickwire remorselessly says that they are on the asteroid, and that there are men, and that therefore there can be no peace while they are there.
Later, Wickwire boards the grounded spaceship and dusts off the three crewmen. The men are dead, embalmed, and frozen at their posts, happily posed as if they were returning to Earth. Returning to his home, Wickwire sits down and shuts off.