Speaking from well worn first hand experience, 86 is a term used in the restaurant and bar industry to denote a patron that has been permanent disbarred from entering the premises, usually due to disorderly behavior. To say that a person has been 86'd is to convey that they have crossed the line and been instructed that their presence in the establishment will no longer be tolerated.
It is also frequently used as terminology in busy commercial kitchens when the supply of a particular item has run out, such as "86 on the scallions!" or writing the item on a posted list of groceries that need to be resupplied. Waitstaff similarly use the term for menu items that are unavailable for the day or for special adjustments to an order, such as "86 the pickle and onions" to keep ingredients off of a sandwich.
Like most slang terms, the etymology of the term "86" is both sloppy and colorful. Stories abound on the background of the word, and nearly every kitchen manager or seasoned bar tender has their own version of how they think the whole thing got started, some of which sound more likely than others.
The greater majority of origin stories in circulation involve alcohol laws and/or prohibition. One very convenient etymology is that there was once a state law in New York called Article 86 that laid out exactly where the line was for a patron to be too intoxicated for a bartender to legally continue serving him drinks. However, precursory research performed by me (i.e. googling) failed to uncover any concrete citations that there was ever any such law. However, this origin makes considerably more sense than most other etymologies of the term.
Other alcohol related origins are related to prohibition, which converges with the first appearance of the term during the 1930's. One colorful, albeit unlikely, story is that the term got started as a local code word at a New York speakeasy called Chumley's. The speakeasy was located at a corner with a main entrance on one street and a side entrance at 86 Bedford Place. When the cops were spotted approaching the speakeasy for a raid or inspection, proprietors and patrons would yelp "86 everybody!" and they everybody would make for the side door at 86 Bedford.
Another restaurant by the name of Delmonico's in New York is often cited as being the home of the term 86. During the late 19th Century, the restaurant supposedly had a numbered menu on which item 86 was the house steak. Item 86 was so popular that it was often out of stock in the kitchen, and the kitchen staff began referring to other items as being similarly "86'd" when they ran out. The kitchen staff took the term to other kitchens through their careers and the term spread throughout the industry. As pleasing as this story may be, it is unlikely to be the true origin for a handful of obvious reasons, including the fact that the term did not actually appear until some 50 years after the item 86 at Delmonico's anecdote is supposed to have taken place.
A decidedly different word history for 86 is that it originated amongst the codes of, not bartenders or surly cooks, but freckle faced soda jerks. Counter jumpers at the soda fountain bars of yesteryear had an extensive coded system of numbers to indicate different items. 33 was a cherry Coke, 19 for a banana split. 99 or 98 meant that manager or assistant manager was coming, and it was time to look sharp. 86 was the number for being out of stock of an item. Another explanation that some bartenders put forward is that 86 was a nonsense codeword that allowed bar and waitstaff to communicate that a customer needed to be removed. They would say 86 as it rhymed with nix, another popular slang term of the early 20th century.
Still, the origin stories of the term 86 are deeply varied, and you'll likely find some completely different version of the etymology by asking any seasoned restaurant or bar worker. One inexplicably morbid explanation is that 86 is the dimensions of a grave: six feet deep and eight feet long. Another history holds that the cauldrons that were used during the Awesome Depression could only hold 85 cups worth soup, and the soup was officially 86'd after that. That particular origin story is an example of what you run across pretty frequently in etymology: people just making shit up.