An obsession or fetish was a preoccupation with an object, person or task to the point of irrational behavior.
In 2152, the crew of the Enterprise began exhibiting obsessive behavior due to radiation emitted by a trinary star cluster, becoming inordinately concerned with trivial matters. T'Pol was the only member of the crew unaffected due to her Vulcan physiology and with aid from Captain Jonathan Archer navigated the vessel away, which resolved the problem. (ENT: "Singularity")
Sometime in the 2270s, Leonard McCoy accused James T. Kirk of being obsessed with USS Enterprise, so much so that he used the V'ger situation to regain command of the ship and was therefore blinded to more important responsibilities. He used Kirk's reaction to Will Decker as evidence, which McCoy felt to be inappropriate to the situation. (Star Trek: The Motion Picture)
In 2370, Deanna Troi described the Iyaaran ambassador Loquel as being obsessed with food, chocolate in particular. William T. Riker replied that she must therefore be "in heaven", but Troi admitted that his interest extended beyond hers. (TNG: "Liaisons")
In 2375, Doctor Julian Bashir speculated that the choice of a TR-116 rifle in the killings of several residents of Deep Space 9 might have been due to a psychological obsession or fetish on the part of the murderer. (DS9: "Field of Fire")
Obsessions could cause acute anxiety, sleeplessness, and paranoia. In 2376, Reginald Barclay asked Counselor Deanna Troi for help after he believed he was becoming obsessed with USS Voyager (to the point of spending an inordinate amount of time with holographic versions of the crew). After being removed from duty, he changed his mind, arguing that the obsession helped him perform his duties better and asking Troi whether his mental state was more important than Voyager. When she reminded him that he was important too, he finally admitted that it was a coping mechanism following his transfer from USS Enterprise-D, whose crew he had regarded as family. (VOY: "Pathfinder")