Alex Rose and Nancy Kendricks are a young yuppie couple who are looking for a perfect home. They finally settle for a grand duplex in Brooklyn, unaware that a seemingly sweet little old Irish lady named Mrs. Connelly is living upstairs.
After having a chat with the old lady (and nibbling on her expired snacks), she tells Alex and Nancy about her late husband Richard, who was a sailor half a century ago. Incidentally, her parrot is named 'Little Dick', after Richard. They bring her a bottle of wine, but she turns it down, claiming that it is a sin "Irish Catholics don't drink?" Even though she drank almost the entire bottle.After Mrs. Connelly goes to sleep early, Alex and Nancy try to get a decent night's sleep, but Connelly listens to the television all night long. Her choice of programing includes Woody Woodpecker, South Park, Hawaii 5-0 and numerous commercials. The next morning, Alex and Nancy haven't had a wink of sleep, and after Nancy goes to work, Alex is plagued by Connelly to do all her unpleasant chores. The next day they go shopping, and Nancy buys a large Peacock ornament while Alex buys an antique typewriter and a Clapmate system. Alex heads upstairs after Connelly leaves to do errands, and once he has installed the Clapmate system, he has to hide in the bathroom when he hears the old woman coming back up the stairs. While she has a bath, he escapes by climbing and then falling out the window and landing in a huge thorn bush.
At night, Alex claps as loud as he can to try and turn Connelly's TV off. However, the old woman figures out what the Clapmate system does and claps it on and off all night, much to Alex' chagrin. Their next attempt is to head up to her apartment and sabotage her home by letting gas seep out of her cooker, putting salt in her sugar bowl, and re-wiring her electrical settings. When all of these misfortunes backfire on Alex and Nancy, they deliberately catch the flu and attempt to pass it on to their annoying neighbor. After informing them about a virus that is going around, Connelly tells them that she has recently had a flu shot.
Desperate, the couple hire a hitman to deal with the old hag and, after selling everything they possess to pay the hitman, wait for the job to be done. Connelly defends herself with her husband's antique harpoon and eventually a fire is started. After experiencing a change of heart, Alex and Nancy rescue Connelly and Little Dick from the fire and call the fire department.
Alex and Nancy finally decide to evict themselves. Just before leaving, they discover that the old woman has died peacefully in her sleep. They contemplate their strange encounters, while it is revealed that the estate agent who sold them the duplex is the old woman's son and the cop who has been constantly carping the couple is his homosexual partner. They and Connelly (who is revealed to not be dead) do this all the time, harassing young couples and faking Connelly's death while collecting a commission.
Although 'Duplex' was renamed 'Our House' in the United Kingdom, it was released on DVD there as 'Duplex.'