The film opens with Charlie Fox (River Phoenix) explaining that his father, Allie Fox (Harrison Ford), is a brilliant inventor with "nine patents, six pending." Allie has grown fed up with the American Dream and American consumerism, believing that Americans "buy junk, sell junk and eat junk," and that there is an impending nuclear war on the horizon as a result of American greed and crime.
Allie and Charlie go to a hardware store to buy components for a new invention, an ice machine known as Fat Boy. Upon seeing that the product was made in Japan, Allie refuses to purchase it. After Allie and Charlie acquire the components at a local dump, he finishes assembling his creation. Allie's boss and asparagus farm owner, Mr. Polski (Dick O'Neill) complains that Allie is not tending to the asparagus, which is rotting. Allie, Charlie, and Allie's youngest son, Jerry (Jadrien Steele), meet Mr. Polski, and Allie shows him "Fat Boy." The machine leaves Polski unimpressed. As he drives past the fields, a dejected Allie comments on immigrants picking asparagus, and says that where they come from, they might think of ice as a luxury. The home of the migrant workers is in a state of disarray, exemplifying their poverty.
That night, Jerry tells "Mother" (Helen Mirren), that he believes something terrible is about to happen. Mother rebuffs her son, explaining that she believes something good will happen. The next morning, Allie throws a party for the immigrant workers before telling his family that they're leaving the United States. After they board a Panamanian barge, the family meets Reverend Spellgood (Andre Gregory), a Christian missionary, his wife (Melanie Boland), and their daughter, Emily (Martha Plimpton). Emily flirts with Charlie. Allie and the Reverend begrudgingly try to get along, despite having entirely different religious views. When the barge docks in Belize City, the families disembark and go their separate ways. Allie, with the consent of the Belize government, purchases a small village called Jeronimo in the rainforest along the river.
Mr. Haddy (Conrad Roberts) takes Allie and his family upriver to Jeronimo. Allie meets the inhabitants and proceeds to start building a new, 'advanced' civilization, in the process inventing many new things. The locals take kindly to Allie and his family, but Allie's will to build a utopic civilization keeps them working to their limits. One day, Reverend Spellgood arrives to convert Jeronimo's citizens. In the process, Allie and Spellgood angrily denounce each other, leading to a permanent schism: Allie believes Spellgood to be a religious zealot; Spellgood believes Allie to be a communist. Allie sets to constructing a huge version of "Fat Boy" that can supply Jeronimo with ice. Upon completing the machine, Allie hears rumors of a native tribe in the mountains that have never seen ice. Allie recruits his two sons to carry a load of ice into the jungle to supply the tribe. Upon arriving, Allie finds that the load has melted, and that the tribe has already been visited by missionaries.
When Allie returns to Jeronimo, he learns that Spellgood has left with much of the populace, scaring them with stories of God's biblical destruction. The near-empty town is visited by rebels, who demand to use Jeronimo as a guerrilla base. Allie and his family accept to accommodate them while Allie constructs a plan to be rid of them. Set on freezing them to death, Allie bunks the rebels up in the giant ice machine, tells Charlie to lock its only other exit, and activates it. The rebels, waking in panic, try to shoot their way through. To Allie's horror, the rebels' gunfire sets off an explosion within the machine. By the next morning, both the machine and the family's home is in ruins. Worse, the chemicals from the destroyed machine have severely polluted the river.
Forced downstream, Allie and his family arrive at the coast. Mother and the children rejoice, believing they can return to the United States. Allie, refusing to believe his dream has been shattered, announces that they have all they need on the beach and, lying, tells the family that America has been destroyed in a nuclear war. Settling on the beach in a houseboat he has built, and refusing assistance from Mr. Haddy, a paranoid Allie believes that the family has accomplished building a utopia. One night, the storm surge from a tropical cyclone nearly forces the family out to sea until Charlie reveals that he has been hiding motor components given to him by Mr. Haddy, allowing them to start the motor on the boat. The family becomes physically and emotionally weaker for lack of food, shelter, and other human companionship.
Traveling upstream once again, the family stumbles across Spellgood's compound. Coming ashore, Allie sees barbed wire, and mutters that the settlement is a Christian concentration camp. While the rest of the family sleeps, Charlie and Jerry sneak over to the Spellgood home. After finding out that the United States was not destroyed and that Emily will assist them in escaping from Allie, Charlie obtains the keys to a jeep. Before Charlie can convince Mother and his sisters to leave, Allie sets Spellgood's church on fire. Spellgood shoots Allie, paralyzing him from the neck down. The family escapes aboard the boat.
The film concludes with the group traveling downriver again, where Allie drifts in and out of consciousness. Allie asks his wife if they are going upstream. She lies to himgoing against the wishes of her husband for the first time. Charlie's final narration reports the death of Allie, but gives hope that the rest of the family can live their lives freely from now on.