The Soviet naval submarine Sprut (which means "octopus" or "kraken" in Russian) draws too close to the New England coast when its captain wants to take a good look at America and runs aground on a sandbar near an island off Cape Cod. Rather than radio in for help and risk an embarrassing international incident, the captain (Theodore Bikel) sends a nine-man landing party headed by his second-in-command Lieutenant Rozanov (Alan Arkin) to find a motor launch to help free the sub from the bar. The men arrive at the house of Walt Whittaker (Carl Reiner), a vacationing playwright from New York City. Whittaker is eager to get his wife Elspeth (Eva Marie Saint) and two children, obnoxious six-year-old Pete (Sheldon Collins) and three-year-old Annie (Cindy Putnam), off the island now that summer is over.
Failing to convince the Whittakers that his group are Norwegians (all of the Russians are conspicuously dressed in sinister all-black clothing), Rozanov draws a gun and promises no harm if the family provides information about military on the island (none) and police force (small), and gives them the keys to their car. Walt and Elspeth provide the answers and the keys, and the Russians depart, leaving behind a young sailor, Alexei Kolchin (John Phillip Law), to guard the Whittakers and, subsequently, their attractive 18-year-old neighbor and babysitter, named Alison Palmer (Andrea Dromm) when she stumbles upon them. The Whittakers' station wagon quickly runs out of gasoline on a rural coastal road, forcing the Russians to walk. They make it to another group of houses where they steal an old sedan from Muriel Everett (Doro Merande), the postmistress; she calls Alice Foss (Tessie O'Shea), the gossipy telephone switchboard operator, and before long, wild rumors throw the entire island into confusion.
As level-headed Police Chief Link Mattocks (Brian Keith) and his bumbling assistant Norman Jonas (Jonathan Winters) try to squelch the inept vigilante movement of blustering Fendall Hawkins (Paul Ford). Meanwhile, Walt and Elspeth manage to overpower Kolchin, who flees, but when Walt and Elspeth along with Pete leave the house to find help, Kolchin returns and takes Alison and Annie hostage. Trying to find the Russians on his own, Walt is re-captured by them. After subduing Mrs. Foss and disabling the island's telephone switchboard, seven of the Russians manage to steal a motorboat and head back to their sub, which is still high and dry. Back at the Whittaker house, Kolchin is by now falling in love with Alison. Walt manages to free himself, and he and Elspeth return to the house and almost shoot Rozanov who arrives there just after they do. With the misunderstandings cleared up, the Whittakers, Rozanov and Kolchin decide to head into town together to clear the air with everyone over just what is going on.
With the rising tide, the Sprut frees itself, and the Russian captain sets out in search of his missing men. He finds Rozanov and Kolchin in the harbor and threatens to blow up the town unless the other seven are returned to him. Chief Mattocks arrives with the rest of the armed villagers who threaten action against the sub's crew on deck with rifles and pistols. As tension mounts, a small boy (Johnny Whitaker) falls from his perch on the church steeple and hangs perilously from a gutter. Forgetting their differences, islanders and Russians unite to form a human pyramid and rescue the child.
With peace and harmony established, the Sprut heads out of the harbor with a convoy of villagers in small boats protecting it. The boat with the seven Russians reaches the group shortly thereafter, and the seven board the submarine, just before two U. S. Air Force F-101B Voodoo jets arrive. They break off after seeing the convoy, and the Sprut is free to sail away to safe waters..