The NBA and NFL drafts are often held in New York City. A draft prospect is sometimes said to be “strong as a bull/ox” (very strong) or someone who can “run like a deer” (very fast). If a player can jump high, it’s sometimes said that he can “jump out of the gym” (or “jump out of a gym”).
 
“Jump out of the gym” is most often said about a basketball player. Basketball is played in gyms and players need to be able to jump to reach the basket. The phrase is not too be taken as literally jumping out of a gym.
 
“Jump out of the gym” has been cited in print since at least 1968. It’s not known if any particular sports person (such as a broadcaster or a coach) coined the phrase.
 
 
Google News Archive
12 July 1968, Sumter (SC) Daily Item, “Gym, Veterans Make Hodgin Optimistic Over Bird’s Future” by Doug Nye, pg. 1B, col. 7:
Others Hodgin will count on are 5-11 Mike Herlott, a junior who can jump out of the gym, 6-1 Fred Brogden and 6-2 Lathan Roddey.
 
Google News Archives
26 January 1970, Hendersonville (NC) Times-News, “‘Wild Bunch” Headed fo H’ville” by Buddy Chapman, pg. 13, col. 6:
“Peterson really likes to go to the boards,” said Carter, “and Black is a boy who can jump out of the gym on a jump shot, the type where he just hangs high in the air.”
 
Google News Archive
3 August 1970, Oxnard (CA) Press-Courier, “OHS Ex-Cager Linked to L.A. Kidnap,” pg. 9, col. 2:
As a basketball player, said Dunn, Swafford was “potentially good. He could jump out of the gym and could move pretty well for a fellow his size.”
 
Google News Archive
24 November 1976, Sarasota (FL) Journal, pg. 8B, col. 1:
“He can jump out of the gym,” said Wilkes. “He’s probably as good as any player in the state.”
     
Sports Illustrated
November 28, 1983
13 Arkansas
Jack McCallum
They’ll be showing Eddie and the Cruisers, Part 3 down in Fayetteville, Ark. this season. Razorback Coach Eddie Sutton likes his “cruiser” types, those big guards who run the floor like deer, jump out of the gym and have the wiry strength to check big people on defense. First it was Sidney Moncrief, then it was Darrell Walker and now it’s senior Alvin Robertson.
 
Google Books
Walt Frazier:
One magic season and a basketball life
By Walt Frazier and Neil Offen
New York, NY: Times Books
1988
Pg. 91:
They know they can all run and jump out of the gym, and so they feel they don’t have to box out the way DeBusschere used to.
   
Google Books
Kobe:
The Story of the NBA’s Rising Young Star Kobe Bryant
By Joe Layden
New York, NY: Harper Paperbacks
1998
Pg. 44:
Yes, he was extraordinarily quick, and he could jump out of the gym. He also had a great outside shot. Still, it wasn’t unreasonable to think he might have difficulty adjusting to the muscular world of the NBA.
   
Google Books
The Smart Take from the Strong:
The Basketball Philosophy of Pete Carril
By Pete Carril and Dan White
Lincoln, NE: University of Nebraska Press
2004
Pg. 34:
Michael Jordan can run like a deer and jump out of the gym; those are the two skills everyone notices, but they totally obscure the fact that he is fundamentally sound.