Seascape opens on a beach. An older couple, Nancy and Charlie, has finished a picnic lunch. As Nancy cleans up, the noise of a jet flying low engulfs the stage. Charlie predicts that a jet will someday smash into a dune.
Nancy expresses her desire to be near the water forever. She loves everything about it and would like to travel from beach to beach. Charlie responds negatively to her dreams. He does not want to do anything or go anywhere.
Charlie's attitude angers Nancy. She points out that life is short. She threatens to have adventures on her own. Charlie's attitude changes after Nancy's outburst, and she retreats from her plan a bit. She is content to enjoy the moment.
After another jet passes by, Nancy reminds Charlie about his childhood desire to live under the water. Charlie tells her how he would sink to the bottom of a pool or lake and sit there until he had to breathe. Nancy encourages him to do this again and get in touch with his youth. Charlie refuses, embarrassed by her insistence.
Nancy changes the subject to their sex life. She tells him about a time in their marriage when she thought of divorcing him. There was tension, and she suspected that he was having an affair. Charlie denies this, and Nancy accepts his word.
Nancy encourages him to sink under the water again and to show her how he did it. Charlie again refuses and turns the conversation to her. He tells her that she was a good wife. Nancy says the same about him, listing the many ways in which he was a good husband. When she is done, she is bitter because the "good life" they have had seems limited to her. Charlie is hurt by her attitude. They argue. Nancy is still angry that his only interest is to rest, while she wants to experience new things.
During a pause at the end of their heated argument, Leslie, a human-sized male lizard, takes a peek at them. Nancy tries to get Charlie to help write postcards, but he declines. Leslie peeks at them again, this time with his female mate, Sarah. Nancy sees the lizards and is intrigued. Charlie is afraid.
Charlie demands that Nancy find him something to defend them with. When she can only find a small, thin stick, Charlie is peeved. Nancy remains interested in the lizards, but when Leslie clears his throat, she fears the lizards might hurt them.
When Leslie waves his large stick, Nancy and Charlie admit their love for each other, Nancy more reluctantly than Charlie. As Leslie and Sarah move forward, another jet flies by and scares them away. Charlie tries to blame the whole episode on bad liver paste sandwiches. He believes they are dead from food poisoning. Nancy ridicules the idea.
Nancy is pleased when Leslie and Sarah return. To protect themselves, Nancy believes they should show submission by lying on their backs with their legs and arms up, as a dog would. Charlie assumes the position, though with great reluctance.