Martial arts were athletic disciplines related to combat for the purpose of recreation or self-defense.
The training of Starfleet personnel in the martial arts was first displayed in Star Trek: The Original Series. During the run of the series, most Human crewmembers would often use martial arts to subdue opponents, leading to the plot device of the "space karate chop" where an adversary could be rendered unconscious by a single blow to the base of the neck. Designed as a quick means to eliminate enemies (and avoid lengthy fight scenes), producers later admitted the concept as unrealistic and "silly", thus leading to more realistic combat scenes in later television series and films. (Inside Star Trek: The Real Story)
As evidenced by the final draft script of ENT: "Hatchery", a Vulcan martial arts maneuver was at one point to have been used in that episode, by T'Pol against a particular MACO. The martial arts move was clearly discarded in favor of stunning the MACO with a phase-pistol.
According to his personnel file in the non-canon video game Star Trek: Starship Creator, Barnaby is a third degree black belt in the art of Tae Kwon Do and lists Earth martial arts among his interests.