In the near future, young children are playing in the bushes of their neat suburban neighborhood. The ringleader, Mink, says she's found what they're looking for and everyone else gathers around. Meanwhile, her mother Mary is working at home and her husband Henry lets her know that he'll be late from work that night. As he goes out to his car, he notices his daughter and her friends playing their new game and calls out a greeting, but Mink waves to him briefly and then goes back to her friends.
Mink and the other children go home for lunch and Mink starts taking out kitchen utensils, much to Mary's dismay and amusement. When she asks what's going on, Mary explains that they're playing a new game, Invasion, and then runs back outside. Two of the neighbor boys, Eddie and Joseph, notice what they're doing. The children have all gathered items from their homes and gather in the bushes to assemble them. Joseph comes over to ask if he can play, but Mink tells him that he and Eddie are too old. Joseph still appears interested, but his friend Eddie taunts him until he rides away.
As she prepares for lunch, Mary watches a newscast about the newly signed earth Mutual Defense Treaty. The nations of the earth have disarmed and are united against any outside attack. Meanwhile, Mink is busy reciting mathematical formulae while her friends write down what she says. She concludes by saying they need a hexagon, and then goes inside. Mink looks through the drawers and finds a cheese grater with a hexagon shape, and then explains to her mother that she has a new friend, Drill. Drill is new in the neighborhood and told Mink to find a hexagon. She goes outside with it and Mary watches, puzzled, as her daughter joins back up with the other neighbor children, who have collected more bits and pieces from around their homes.
When Mary calls Mink into her lunch, her daughter plays with a top that spins on its own. Mary insists that she stay for lunch, but Mink tells her that the end of the world is coming and she needs to go play with Drill. Mary forces her to explain and Mink says that Drill is from another planet and is playing a game where his people are going to invade Earth. They recruited the younger children of Earth to help them. Drill is from the fourth dimension, and relies on children's imagination to help him. When Mink informs her mother that Drill promised that they wouldn't have to take baths and could stay up as late as they want, Mary wants to know who Drill's parents are so she can talk to them. Mink notes that Drill said that parents would try to interfere, and starts to go. As she leaves, she promises Mary that when the invasion comes, she won't be hurt... much.
As the children play their game, Eddie and Joseph come back. Eddie makes fun of them again, and Mary tells him that when the invasion begins, he'll be killed. Angry, Eddie insists that Joseph leave with him and the two boys go.
Mary is working and receives a videophone call from her sister Helen on the West Coast. As they talk about Mink and her new game, Mary notices that her nephew Eddie is also looking for a hexagon-shaped object that he says is for his friend, Drill. Mary begins to realize something sinister is going on, and hears a scream from outside. She goes out and sees one of Mink's friends running away. When Mary demands an explanation, Mink tells her that the girl is too old now and got scared. She tells her mother that 5 o'clock is "zero hour," and that Drill is stuck halfway through to Earth. They have to use the items they've assembled to bring him the rest of the way through so he and his people can begin the invasion. Mary takes out the top from earlier, holds it up, and Mary watches it spin and then disappear. Frightened, Mary tells Mink to come home by the time her father returns from work and runs inside. The children turn back to their device, which glows with an unearthly energy.
Inside, Mary watches the clock and waits for Henry to arrive. He returns at 4:55 and sees Mink playing with the other children. She tells him to go inside and he enters the house and pours himself a drink. Mary guilty wonders if children ever forget how parents dominate them, and then jumps as she hears a distorted electrical noise from outside. Henry doesn't understand why she's so concerned, but Mary begs him to stop the children's Zero Hour game. Before he can do anything, they hear another electrical noise and a man screams. Mary panics and runs up to the attic, and Henry follows her, trying to determine what's wrong.
In the attic, Mary lures Henry in after her, locks the door, and then throws the key into the storage boxes. She breaks into tears and tells her husband that they can sneak out after midnight, and tells him to keep quiet. Henry insists on finding the key and dismisses he concerns. The phone downstairs rings and Mary figures that it's her sister, trying to warn her. Henry finds the key and starts to unlock the door, and hears someone moving around downstairs. Ignoring Mary, he calls out and Mink hears him. As she comes upstairs, Henry begins to realize that something is wrong. He locks the door and hides in the corner with Mary. Mink knocks on the door and asks if they're inside, as an unearthly blue glow filters around the cracks. When her parents refuse to respond, the door dissolves and Mink peers in. She sees her parents and says "Peekaboo," while behind her, alien blue figures arrive to complete the game.