The story
Attested 1950s. Presumably from a bawdy military joke, featuring a barrel with a glory hole, whose punchline is “It’s your turn in the barrel.” The joke has many variants and embellishments, but the kernel is: a new sailor is being given a tour of the boat, and is shown a barrel with a glory hole, which they are able to use any time, except Tuesdays. When he asks why not Tuesdays, he is told: “Because it’s your turn in the barrel.” Other hypotheses is that it is derived from like shooting fish in a barrel, or that it referred to a form of punishment in which the malefactor was confined within a barrel.
Etymology from Wiktionary, CC BY-SA 4.0.