The cat story gets me every time. A mature, well-to-do husband and father sits in his manorlike suburban living room, drink in hand, and recalls a pet cat he had in his youth who inexplicably turned against him. Nothing violent. The cat just stopped liking him and wanting to be in the same room with him. Not long after that, he had the animal euthanized; now he is haunted by the experience.

“A Delicate Balance,” Edward Albee’s Pulitzer Prize-winning masterpiece and the grand current production at the Berlind Theater of the McCarter Theater Center, has been described more than once as a horror story. There are no literal ghosts or monsters. The story, directed here by Emily Mann, the theater’s artistic director, simply forces anyone paying attention to acknowledge the paralyzing fear that simmers within every waking second of human life and the awful fragility of connections.

If that terror can break through the privilege of wealthy, well-educated people like Agnes (Kathleen Chalfant) and her husband, Tobias (John Glover), the man with the cat, then clearly no one is safe.

The event that upsets the couple’s equilibrium is the unexpected after-dinner arrival of Harry (James A. Stephens) and Edna (Roberta Maxwell). All four try to behave as if dropping by without having called is not a faux pas, but that lapse in manners is nothing compared with the reason for it. Harry and Edna were at home when both were stricken by an unnamable terror, and they have fled, coming for sanctuary to their very best friends in the world.