The Enterprise is docked at the Remlar Array in preparation for a baryon sweep, a space-age decontamination procedure using a deadly beam that sweeps across the entire ship, starting from the aft and moving towards Ten Forward. To this end the entire ship needs to be evacuated and all systems shut down.
Picard and other members of the crew are invited to a cocktail party by Commander Calvin "Hutch" Hutchinson, an officer who is notorious for his ability to engage in small talk. Cornered by Hutch, Picard sees an opportunity to escape the party by going horseback riding nearby, and "urgently" ducks out to retrieve his saddle from the Enterprise before the decontamination gets underway. On his way back to the transporter room he notices that some panels in one of the corridors have been opened. As he investigates he is approached by a technician from the array (Tim Russ, who later portrayed Tuvok in Star Trek: Voyager) who tries to explain the purpose for removing the panels. However Picard does not believe him. The technician attacks with a laser welder but Picard, ironically, uses a Vulcan Neck Pinch to subdue him.
Meanwhile at the cocktail party, Data provides comic relief as he learns small talk from the smarmy and garrulous Commander Hutchinson (David Spielberg). The two excitedly trade trivialities throughout the cocktail party ("...how long can two people talk about nothing," exclaims an exasperated Riker). However, Geordi's visor picks up some strange readings coming from the centerpiece on a table. When he investigates, the hosts of the party - who have become concerned that their involvement in the plot unfolding on the Enterprise will be exposed - decide to take the Enterprise senior staff hostage, injuring Geordi and killing Commander Hutchinson in the process. The rest of the episode cuts back and forth between the crew's plan to neutralize their captors and Picard's adventure on the Enterprise.
With the ship's computer shut down, Picard is trapped and unable to transport away. He is eventually captured but claims to be Mot, the ship's barber. He is moved to engineering, where he observes the mercenaries and learns that they are planning to steal particles from the Enterprise's warp reactor, which they will sell to another group that wants to build a powerful explosive weapon. He manages to cause an emergency in engineering, resulting in the isolation doors closing and distracting the mercenaries long enough for him to escape. He then engages in a ruthless game of cat-and-mouse, acquiring a crossbow from Worf's quarters and using his intimate knowledge of the Enterprise to outwit the team of mercenaries. Although Picard spared the technician's life in their first encounter, he subsequently takes no prisoners: he either kills, or allows the demise of, every mercenary before the Enterprise crew's plot succeeds and Data rescues him at the last minute as the baryon sweep comes toward him.
The last segment in sickbay contains an oblique dialog reference to the old television sitcom "Mister Ed". When Worf returns Picard's saddle, they exchange words reminiscent of the words to Mister Ed's theme song. Picard ends his line with, ".... a horse," to which Worf replies, "of course."