The story is set in the year 2055 and opens by introducing a company called “Time Safari, Inc.,” with a brief description of its unique services. The company offers people the chance to travel back in time to hunt animals: “We take you there. You shoot it.”

Eckels, a nervous but ambitious hunter, is buying a ticket for ten thousand dollars. He is concerned about safety and learns that survival is not guaranteed. He is introduced to Mr. Travis, his “Safari Guide in the Past,” and learns that Travis determines what he is permitted to shoot and that any disobedience can result in a fine and potential government involvement.

Eckels admires the time machine and strikes up a conversation with the man behind the desk. They discuss the results of a recent election for President of the United States. Both Eckels and the man rejoice at the fact that their favorite candidate, Keith, won the election instead of Deutscher, who is an authoritarian. The man tells Eckels that they received several requests from people who asked about traveling back in time to escape, had Deutscher won. He explains that the company’s purpose is to send people on hunting trips, not to help them escape to the past. The man tells Eckels that all he needs to worry about now is shooting a Tyrannosaurus Rex. He hands him a release form, indicating that if he signs it, the company will not be responsible for anything that happens to him.

Angrily, Eckels asks if he is trying to frighten him. The man answers, “Frankly, yes. We don’t want anyone going who’ll panic at the first shot.” He tells Eckels that six guides have died in the last year, along with a dozen hunters. He offers Eckels a chance to tear up his check and give up. Eckels considers it, but he decides to continue.

The group journeying to the past includes Eckels, two other hunters—Billings and Kramer—and Travis and his assistant, Lesperance. They enter the time machine and travel back 60 million years. During the journey, Eckels nervously moves in his seat and tightly grasps his gun. He tries to put on a brave face, asking whether their guns can kill a dinosaur. Travis explains that the guns work if the hunter hits the dinosaurs correctly. Some dinosaurs have two brains and are difficult to kill. Travis advises the hunters to stay away from such dinosaurs. Travis goes on to explain in graphic detail the correct method of killing a dinosaur. He tells the hunters to “[p]ut your first two shots into the eyes, if you can, blind them, and then go back into the brain.”

The machine howls as Eckels reflects on how every past hunter would envy them today.