The series follows the adventures of the crew of the Federation starship Enterprise commanded by Captain Jean-Luc Picard. In this episode, First Officer Riker leads the Enterprise in dealing with a strange new alien race called the Pakleds while Captain Picard undergoes artificial heart replacement surgery at a starbase.
Captain Jean-Luc Picard is due for a medical operation: his artificial heart is starting to break down and he needs a replacement. Picard refuses to have the Chief Medical Officer, Doctor Pulaski, do it, and preferably not on board the "Enterprise" because he is concerned about his image with the crew, so she convinces him to go to a starbase for the operation. He announces that he will be joining young Acting Ensign Wesley Crusher, who is on his way to that base for Starfleet Academy entrance tests, in a voyage via shuttlecraft. During the trip, the relationship between the two grows, Picard shares a bit of his past, and directs Crusher to some classic books (including William James).
Meanwhile, the Enterprise encounters the Mondor, a ship of strange aliens. These aliens, called Pakleds, request the Enterprise's help to fix their broken ship. Judging from the way the aliens discuss the problem, the crew thinks they are so ignorant, it's strange they were able to build and fly the ship in the first place. The Enterprise agrees to send Chief Engineer La Forge over to fix the problem.
In the Pakled ship, La Forge finds out that the Pakleds are technologically and linguistically unsophisticated but not entirely helpless. Their modus operandi is to exaggerate their distress and helplessness to lure in "good Samaritans" whose advanced technology and knowledge can then be taken by force in a surprise attack or kidnapping. The Pakleds use this ruse to take La Forge hostage, hoping to have him make them more weapons.
Meanwhile, Picard's operation is going wrong and he is about to die. The doctors summon the only specialist more knowledgeable about the procedure than they, Dr. Pulaski.
On the Enterprise, Commander Riker and Lt. Worf covertly establish a plot with La Forge using code in their communication to amaze the Pakleds into thinking they are no match for the Enterprise. The ruse, featuring a spectacular light show of hydrogen gas passed through a forcefield, works and the Pakleds release La Forge. The ship hurries to the starbase so that Pulaski can save Picard's life.
The episode ends with Picard waking up, looking into the smiling face of Dr. Pulaski, who informs her captain the operation went well, only to get a weak-voiced disgruntled yet sincere thanks before leaving him to recuperate, slightly embarrassed as he realizes that the whole crew must know of his predicament per consequence.