Twelve-year-old Billy Gillfoyle rides with his father, Bill, along a winding country road to Lake Katrine. There, Billy will spend the summer with his mother, Kim, a math professor, while Bill returns to New York City. Their separation troubles Billy, who feels powerless to change it.

At the cabin, Kim mentions a dock, a rowboat, and an island, and notes that some neighboring boys appear to be Billy’s age. Billy notices security cameras mounted throughout the property. Kim explains she rented cheaply from Dr. Xiang Libris, a colleague who owns the island. The low rent reminds Billy of his parents’ arguments about money, and he briefly imagines winning the lottery to bring them back together. Bill waves goodbye and drives off.

Kim reminisces briefly about her past trips to Lake Katrine with Bill and then shows Billy a cupboard stocked with his favorite peanut butter crackers before retreating upstairs to work on her dissertation, a project that will keep her busy all summer.

Billy finds a security camera in his bedroom and covers the lens with his bathing suit. Outside, he identifies his two neighbors’ properties, a sleek glass house on one side and a mismatched two-story structure featuring a tower and a circus tent on the other. Behind the cabin, he finds a large satellite dish. He sees a red rowboat at the dock and an island far across the lake. Suddenly, a voice calls for help from the rambling house next door.

Billy finds five-year-old Alyssa Andrews, whose doll became stuck in a tree after she threw it out a window. Her brother, Walter, isn’t allowed to climb trees due to his asthma. Billy climbs the tree and frees the doll, but he accidentally drops his iPhone, which shatters on a rock below. Knowing that asking his mother to buy a replacement is out of the question, Billy accepts the loss. When Alyssa insists he hug the rescued doll, he reluctantly obliges just as “three tough-looking guys” on bicycles roll up and see him doing so.

The lead biker mocks Billy for holding a doll and gives him an insulting nickname, “Weedpole.” Alyssa identifies the boy as Nick Farkas. Farkas and his companions make fun of Billy’s last name. When Billy spots Space Lizard comics in Farkas’s basket and tries to bond over their shared interest, Farkas responds with threats. He points to the glass house, confirming he’s Billy’s neighbor on the other side, and tells Billy to stay away.

Dr. Libris reports that Billy is in the cabin and notes an issue with the camera in the boy’s bedroom. He’s pleased the iPhone broke, reasoning that without digital entertainment Billy will turn to the books inside the cabin. Libris predicts Billy will soon pass a final aptitude test by finding a hidden key, after which he says first contact will occur quickly.

After dinner, Billy notices a picture of people walking on impossible staircases in the living room. His mother explains that it was made by the artist M. C. Escher and mentions her own work on parallel universes. She invites him to read the books in Dr. Libris’s study and notes that a large bookcase inside is locked, so he’ll need to find the key.

The windowless study is lined floor-to-ceiling with bookshelves and has a security camera above the door. Billy spots a The Wizard of Oz cuckoo clock with its hands frozen at seven and twelve. Sitting in a leather reading chair, he finds a hidden switch under the cushion. On the third click, small spotlights illuminate a massive ornate bookcase covered in 3-D wood carvings of storybook characters. Inside the locked glass doors, The Labors of Hercules sits open on a stand.

Billy finds a tiny slip of paper taped behind the lion-shaped brass keyhole and retrieves a toy magnifying glass from in front of a volume of Sherlock Holmes stories. The paper contains a riddle asking what odd number becomes even when one is taken away. Billy determines that removing the letter “s” from “seven” leaves “even.”

He locates The Seven Voyages of Sinbad the Sailor but finds no key inside. He then notices that the keyhole is shaped like the Cowardly Lion and that the frozen cuckoo clock reads seven o’clock. Opening the clock’s small cuckoo door, he finds an antique skeleton key with the Wizard’s face on it. It fits the bookcase lock.

Dr. Libris notes that Billy solved the aptitude test faster than he anticipated, which makes him confident that Billy is key to the project’s “extraordinary future.”

Billy opens the bookcase and picks up The Labors of Hercules and is amused by the bookplate’s pun, “Ex Libris X. Libris.” He reads about Hercules wrestling the giant Antaeus, who grows stronger every time he touches the ground. Billy becomes so absorbed in the story that Antaeus’s voice seems loud enough to rattle the bookcase glass. He then realizes the voice seems to be coming from outside, far in the distance.

Billy closes the book. When he reopens it, another shout and rattle follow. He asks his mother whether she heard yelling, but she heard nothing. The voices resume, but they’re still inaudible to Kim. On the back porch, Billy hears both Hercules and Antaeus calling from the island’s direction and spots a giant shadowy figure moving along its shore.

Back in the study, Billy closes the book, and the strange phenomena stop. He locks the book away, puts the key back in the clock, and waits. After confirming that the figure on the island is gone, Billy treats the situation as a new riddle to solve.

Monitoring instruments show Billy’s theta brain wave numbers are well beyond anything Dr. Libris has previously recorded. Libris calls Billy’s mind extraordinary and hopes his imagination will produce significant financial rewards for the doctor and his team.

The next morning, Billy concludes the sounds stopped because the book was closed. He tells his mother he wants to keep reading. Kim recommends Robin Hood and explains that Maid Marian was just as capable as Robin before returning to her research on alternate realities.

Billy retrieves the key, takes Robin Hood from the bookcase, and reads a scene in which Robin and Maid Marian confront a bounty hunter. From the direction of the island, he hears swords clashing, Antaeus’s roar, and Robin Hood and Marian’s voices.

Alyssa’s arrival interrupts him. He closes the book, and the sounds stop. Alyssa mentions Walter just read her The Three Billy Goats Gruff, prompting Billy to wonder what would happen if those characters appeared on the island, too.

Billy locks Robin Hood away and keeps the key in his pocket. He convinces himself the whole thing is an elaborate prank using actors, cameras, and sound effects, but he can’t explain how a figure at least 15 feet tall was created or how Libris would know which book Billy was reading. By midday, he resolves to row to the island. He tells his mother, promises to wear a life jacket, then lies and claims he knows how to row when she asks.

At the dock, Farkas and his friends arrive on a Jet Ski, boast about a new Space Lizard game, and threaten to return and sink Billy’s rowboat. As they head inside, Billy hears Hercules shout from the island while Farkas and his friends show no reaction.

Billy struggles into the rowboat and spins in circles before something large guides the hull from below. Looking into the water, he sees Poseidon, who pushes the boat to shore with his trident. Billy reaches the island in minutes. Following a path into the forest, he discovers that a massive wire-mesh dome covers the entire island, accounting for its hazy appearance from the cabin. He passes through an opening cut in the netting and presses on, still convinced that everything he’s experienced is the work of actors and special effects.

Billy reaches large, locked wrought-iron gates decorated with metal carvings similar to those on the bookcase. A powerfully built, battered man in a lion-skin cape appears on the far side and begs for help. Billy realizes the man is Hercules. He explains his quest to collect golden apples for King Eurystheus and his inability to defeat Antaeus. Before they can make sense of the situation, the ground shakes and Antaeus, the 15-foot rock creature Billy saw earlier, bursts from the forest and seizes Hercules in a crushing hold.

Billy is now certain that Antaeus is not a puppet or performer. Hercules pleads for help. Billy considers fleeing but can’t abandon Hercules or leave the mystery unresolved. Remembering the bookcase key in his pocket, he tries it on the gate. The gates spring open, and Billy steps into the clearing.

Billy tries to intimidate Antaeus by invoking Zeus, but the ground splits open, and a voice introduces herself as Gaia, goddess of earth and Antaeus’s mother. She confirms that Antaeus draws strength from contact with her, which is why throwing him down only makes him stronger. Gaia threatens that Billy will be Antaeus’s next target before vanishing back into the earth.

Billy distracts Antaeus with a fabricated story about a belated Mother’s Day gift, persuading the giant to set Hercules down. He then instructs Hercules to lift Antaeus without letting his feet touch the ground and hurl him into the lake. Hercules obeys, and a massive splash signals the giant’s defeat.

The quiet is short-lived. Robin Hood and Maid Marian step out of the forest and, mistaking Billy for an agent of the Sheriff of Nottingham, draw their swords and threaten him.