A barstool was a type of chair designed for seating in a bar.
Quark's on the Promenade of Deep Space 9 had many barstools.
In 2372, Elim Garak added extra padding to the seat of Morn's new trousers, saying it would make swiveling on a barstool more comfortable. (DS9: "Body Parts")
In 2373, Morn hit Quark with a barstool after Quark suggested that the space station would likely be the Dominion's first target in a full-scale assault against the Federation and everyone on board would likely perish. After attacking Quark, Morn screamed "We're all doomed!", stripped nude, and ran to the Bajoran temple to pray to the Prophets for protection. (DS9: "Blaze of Glory")
In 2374, Quark installed new backless barstools, much to the frustration of Constable Odo, who ordered him to revert to the old ones by the end of the day, citing Station regulation twenty five sixty two, paragraph four. Later, he complied unwillingly, accusing the Changeling of stifling his creativity, which caused Odo to remark that "Only you would consider barstools to be a form of artistic expression.". (DS9: "The Sound of Her Voice")
While educating Seven of Nine about romance in 2375, The Doctor asked her to "pull up a barstool and pretend you've come here to meet the man of your dreams." (VOY: "Someone to Watch Over Me")
In 2380, the command crew of the USS Cerritos had a meeting about choosing new chairs for the conference room. Commander Jack Ransom advocated for barstools, calling them "cool", but Lieutenant Shaxs said that barstools hurt his back. (LD: "Moist Vessel")