Picard's mentor, professor of archaeology Richard Galen (Normal Lloyd), comes aboard the Enterprise and informs the captain that he is on the cusp of completing an archaeological discovery that has profound galactic meaning, though he can't explain exactly what that means. Galen asks Picard to accompany him on the final leg of this quest, but to do so would require Picard to take a year or more off from Starfleet, essentially meaning the end of his command of the Enterprise. (Supposedly, this discovery would happen much faster if Starfleet would dedicate a ship to it, which begs the question of why archaeological discoveries of such alleged profound meaning aren't given adequate resources by the science-centric Starfleet.) Picard declines the invitation, and Galen's response to Picard's refusal is an excessively harsh guilt trip, to say the least.
Not long after, Galen's chartered ship is attacked and Galen is killed, leading Picard to take up the mantle of the cause under that classic motivator, They Killed My Mentor. From here, Picard and the crew go to work on solving the mystery, which Galen had discovered were carefully hidden pieces of an ancient computer program, whose pieces came from encoded DNA patterns from various worlds scattered across the quadrant — a sort of interstellar Da Vinci Code, if you will.
"The Chase" is probably best viewed as an Indiana Jones adventure (minus the action and stunt sequences) employing starships to track down ancient DNA fragments rather than employing planes, boats, and tanks to track down ancient religious artifacts. Instead of globetrotting through Europe and Asia, we warp from star system to star system. Like an Indy adventure, the story hints at a discovery of the utmost profundity while keeping the action focused on moving unpretentiously from A to B; "The Chase" is probably the right title for what this is.
The labyrinthine plot details defy synopsis because (1) I didn't write them down when I watched it, and (2) to describe it in detail is to futilely reduce the story to who goes where and when and with whom while trying to assemble a puzzle that is built from (1) DNA fragments transcribed into a digital code and (2) plenty of exposition. Suffice it to say the Klingons, Cardassians, and Romulans all become involved in the chase, because everyone has their own piece of the puzzle. Given that the puzzle has been waiting for billions of years to be solved, it seems awfully convenient that all pieces by all parties just happen to be in play within the same roughly three-day period. (There might have been a rationale that explained the coincidental timing, but I neither recall nor care what it was.)
The solution requires all the different races to work together to assemble the puzzle to activate the program, which turns out to be a holographic message recorded billions of years ago by a society that seeded the planets throughout the galaxy with their building blocks of life from which all humanoids evolved — which explains why there are so many similar humanoid lifeforms in Star Trek, you see. Their message: We live on in all of you, and you all share something genetically in common with each other. It's a Trekkian message if I've ever heard one. I don't know if this revelation works as science fiction, but it works as an intriguing payoff to an adventure that sustains the hour but will never be mentioned afterward, despite its jaw-dropping implications.
I like but do not revere "The Chase." It is an effective adventure yarn that has a revelation that strives to appear astounding while it's unfolding and yet almost wants to be casually dismissed afterward. You know this is not meant to be taken all that seriously when the Klingon hears the ancient revelation and says: "That's all? If she were not dead, I would kill her!" You almost sense the writers were hedging their bets by including that line after showing their hand. Believe this tale, they seem to say — or if not, then not.