The colloquial phrase to put one’s cue in the rack means: to give up, to retire, also, occasionally, to die.

The image is, of course, of a billiard-player putting the cue back in the rack when the game is over.—Cf. the synonymous phrase to hang up one’s fiddle, based on a similar image, that of a fiddler who has finished playing.

The earliest occurrences of the phrase to put one’s cue in the rack that I have found are as follows, in chronological order:

1-: From The Boston Post (Boston, Massachusetts, USA) of Monday 10^th June 1895:

It is rumored that Mr. Oddie of Brooklyn was at the Quincy last week. Until last year, when Mr. Oddie put his cue back in the rack, he had been the amateur billiard champion of his city for four or five years. But his cue must be in the rack to stay, for no reports of an Ives touch have come from the Brattle street green tables during his stay.

2-: From The Daily Herald (Brownsville, Texas, USA) of Tuesday 26^th January 1897—Jacob Schaefer (1855-1910) was a U.S. billiard-player:

SHAEFER’S WRIST WEAK
The Wizard Will Provably [sic] Have To Put His Cue in the Rack.

St. Louis, Mo., Jan. 22.—In all probability Jacob Shaefer, the expert billiardist of America will soon have to retire as a professional. His wrist, which was severely sprained by a fall from a street car in Chicago last winter, is the cause.

3-: From The Scottish Referee: A Record and Review of Outdoor Recreation (Glasgow, Lanarkshire, Scotland) of Monday 24^th July 1899—John Roberts (1847-1919) was a British billiard-player:

A Loss to Billiards.

All devotees of the green baize will be sorry to hear that Champion John Roberts contemplates retiral at the end of the incoming season owing to failing eyesight. Roberts consulted a well-known specialist, who informed him that the growing weakness is not the result of disease, but of the ravages of time. The champion has, therefore, made up his mind to put his cue on the rack for evermore.

4 & 5-: From the column The Man Higher Up, published in The World (New York City, New York, USA):

4-: Of Tuesday 9^th February 1904:

“I see,” said the Cigar Store man, “that John D. Rockefeller has retired.”
“Yes,” replied the Man Higher Up; “but what has he retired to? The ante-room of the receiving vault for his. After accumulating the biggest fortune a man ever grabbed in a lifetime, he takes his ball out of the game and puts his cue in the rack to enjoy himself, but he hasn’t got any stomach and his physician sleeps with his shoes and pants on a framework at the side of the bed, like a fireman, waiting for a call to the Rockefeller mansion any minute.”

5-: Of Monday 28^th March 1904:

“I see,” said the Cigar Store Man, “that Gov. Odell will probably resign late in the summer.”
“If he takes charge of the machine to run it during the campaign he certainly ought to put his statesman cue in the rack,” replied the Man Higher Up.

6-: From The Evansville Courier (Evansville, Indiana, USA) of Wednesday 9^th March 1910:

JAKE SCHAEFER PUTS HIS CUE IN THE RACK
Greatest Billiard Player in History Dies in Denver of Tuberculosis

7 & 8-: From two obituaries of John Roberts:

7-: From an obituary by ‘Snooker’, published in The Referee: A Journal of Sport, Pastime and the Stage (Sydney, New South Wales, Australia) of Wednesday 31^st December 1919:

The most remarkable personage in the billiard world has put his cue in the rack.

8-: From an obituary by ‘Jigger’, published in the Darling Downs Gazette (Toowoomba, Queensland, Australia) of Saturday 3^rd January 1920:

The doyen of the billiard world, Mr. John Roberts, has put his cue in the rack for the last time, his death being announced by cable during the week.

9-: From The Referee: A Journal of Sport, Pastime and the Stage (Sydney, New South Wales, Australia) of Wednesday 20^th July 1921:

BILLIARDS
CHARLES DAWSON PUTS HIS CUE IN THE RACK
[…]
(By SNOOKER)

The death of Charles Dawson removes one of the great players from the billiards world. The little Yorkshireman, though not quite rising to the heights of John Roberts, in the all-round game, or W. J. Peall at spot billiards, might be classed as the next best in the list of great English billiardists.

10-: From Secrets of the Circus, by Albert Geyer, “world’s champion tumbler, retired”, published in The Los Angeles Times (Los Angeles, California, USA) of Sunday 4^th March 1923:

“I concluded I was playing the wrong game doing trapeze work, so I put my cue in the rack and took up clowning.”