“The bases are F.O.B. (full of Brooklyns)” was a catchphrase of Brooklyn Dodger announcer Red Barber (1908-1992), meaning that there were Dodgers standing on first base, second base and third base. “FOB” is an established shipping term that means “free on board” (also, “freight on board”). “F. O. B.—full of Brooklyns” has been cited in print since at least 1940.
 
Red Barber also used “F.O.B.” to indicate that the bases were full of Bucs (Pittsburgh Pirates)  or Birds (St. Louis Cardinals).
 
   
Wikipedia: Red Barber
Walter Lanier “Red” Barber (February 17, 1908 – October 22, 1992) was an American sportscaster.
 
Barber, nicknamed “The Ol’ Redhead”, was primarily identified with radio broadcasts of Major League Baseball, calling play-by-play across four decades with the Cincinnati Reds (1934–38), Brooklyn Dodgers (1939–1953), and New York Yankees (1954–1966). Like his fellow sports pioneer Mel Allen, Barber also gained a niche calling college and professional football in his primary market of New York City.
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Brooklyn Dodgers
Barber had been hired by Larry MacPhail, then president of the Reds. When MacPhail moved on to be president of the Dodgers for the 1939 season, he took the play-by-play man along. In Brooklyn, Barber became an institution, widely admired for his folksy style. He was also appreciated by people concerned about Brooklyn’s reputation as a land of “dees” and “dems”.
 
Barber became famous for his signature catchphrases, including these:
 
“They’re tearin’ up the pea patch” – used for a team on a winning streak.
“The bases are F.O.B. (full of Brooklyns)” – indicating the Dodgers had loaded the bases.
 
Wikipedia: FOB (shipping)
FOB is an initialism which pertains to the shipping of goods. Depending on specific usage, it may stand for Free On Board or Freight On Board. FOB specifies which party (buyer or seller) pays for which shipment and loading costs, and/or where responsibility for the goods is transferred. The last distinction is important for determining liability for goods lost or damaged in transit from the seller to the buyer.
     
26 April 1940, Daily News (New York, NY), “The Power House” by Jimmy Powers, pg. 54, col. 1:
Add Cracks: Red Barber’s “The bases are now f. o. b., full of Brooklyns.”
 
14 May 1940, Brooklyn (NY) Eagle, “Jo Ranson’s Radio Dial Log: ‘Red’ Barber’s Lingo Gets an Analysis,” pg. 10, col. 6:
“F. O. B.”: Bases “full of Brooklyns.”
 
10 August 1940, Daily News (New York, NY), “Listening In” by Sid Shalit, pg. 23, col. 6:
Doing a satisfactory reportorial job on the unpredictable Brooklyn Dodgers is a man-sized job. WOR’s Red Barber himself, has become as colorful as the team he covers, particularly his unique vocabulary. Sample Barberisms: “The boys are tearing up the pea patch,” meaning the club is red hot; “In the catbird seat,” meaning the Durochermen are out in front, sitting pretty; and the soul-stirring rallying cry along the Gowanus—“F. O. B.,” meaniung the bases are loaded with run-hungry Dodgers.
 
5 April 1944, Trenton (NJ) Evening Times, “Olmo Leads Dodgers to 6-4 Win Over Yanks Here,”  pg. 14, col. 1:
Olmo stepped to the Dunn Field plate in the third inning with the bases F. O. B.—full of Brooklyns.
 
Old Fulton NY Post Cards
23 July 1947, New York (NY) Post, “Pet Baseball Cliches Pesty” by Paul Denis, pg. 44, col. 2:
The latter’s (Red Barber’s—ed.) favorite byword is, “The bases are F.O.B.”—full of Brooklyns, or full of Bucs (Pirates), or full of Birds (Cardinals).
 
2 September 1949, Omaha (NE) World Herald, “Red Smith’s Views of Sport,” pg. 23, col. 1:
What is saddest of all, young baseball writers now are borrowing from the language of radio. Today you can read, if your stomach can stand it, that there were “ducks on the pond” or that the “bases were F. O. B. (full of Brooklyns).”
   
Google Books
Sports on New York Radio:
A play-by-play history
By David J. Halberstam
Lincolnwood, IL: Masters Press
1999  
Pg. 232 Red (Barber—ed.) introduced his New York constituency to homespun Southern idioms never heard before in the big metropolis. Newsweek printed just a few of them in 1945:
Sitting in the catbird seat........everything is going your way
I’ll be a suck-egg mule...........Red is pretty concerned
A can ‘a corn........................an easy-to-catch fly ball
F.O.B.................................the bases are full of Brooklyns
The bottom of the pickle vat....the Bums are in bad trouble
 
Google Books
The Dickson Baseball Dictionary
By Paul Dickson
New York, NY: W.W. Norton & Co.
2009
Pg. 334:
FOB 1. hist. An initialism from the days of the Brooklyn Dodgers for bases that were “full of Brooklyns” or “full of Bums.” The term was coined by Dodgers broadcaster Red Barber. 2. An initialism used in Pittsburgh for bases that were “full of Bucs.” The term was used by Pirates announcer Rosey Rowswell. 3. An initialism used by BaltimoreOrioles announcers for “full of Birds.”