The title is a double play on words. One is the typical pun between "hare" and "hair", with the bunny (who was already grey-haired) rendered "old and grey" for this cartoon. The title also refers to the old song, "The Old Gray Mare". Some theater cards for this cartoon gave the alternate spelling, The Old Gray Hare.
Elmer sits under a tree, crying over never being able to catch Bugs. A voice tells Elmer that he would eventually catch him, and proceeds to transport him "far into the future" past the years 1950, 1960, 1970, etc., until reaching the then-distant year of 2000.
This offers the chance to use some contemporary gags with a futuristic twist, as Elmer finds a year 2000 newspaper. One headline says, Smellevision Replaces Television! Carl Stalling Sez It Will Never Work!" In sporting news, "Bing Crosby's Horse Hasn't Come In Yet!" (Crosby was known for investing in racehorses that did poorly).
By now, both Elmer and Bugs are very old and wrinkled ("What's up, prune-face?") - Bugs even has a large white beard and a cane - and lumbago - but their chase resumes. This time Elmer is armed with a "Buck Rogers" ray gun. After a short chase (at slow speed, due to their ages), Elmer gets the upper hand, shooting Bugs with his ultra-modern weapon.
At the moment when it seems Elmer has finally beaten his nemesis, the apparently dying Bugs thinks back to when he and Elmer were much younger. This leads to a flashback sequence with a baby Elmer hunting a baby Bugs (both are still in diapers; Bugs, whose "baby" voice is virtually identical to the normal voice of Blanc's Tweety, is drinking carrot juice from a baby bottle; Elmer is crawling and toting a pop-gun; and they interrupt their chase to take a baby nap-time together.)
After the flashback is over, a tearful Bugs starts to dig his own grave, with Elmer getting equally emotional. Just at the point where it seems that Bugs is going to bury himself, he switches places with the weeping and distracted Elmer, and cheerfully buries him alive instead ("So long, Methuselah!") The buried Elmer quips, "Weww [well] anyway, that pesky wabbit is out of my wife [life] fowevew [forever] and evew [ever]!" However, Bugs suddenly pops in and repeats the popular catchphrase of the "Richard Q. Peavey" character from The Great Gildersleeve, "Well, now, I wouldn't say that," plants a kiss on Elmer, then hands him a large firecracker with a lit fuse, and quickly departs. While Elmer shivers and doesn't do anything, the screen immediately fades out and Robert Clampett's famous vocalized "Bay-woop!" is heard with the firecracker still hissing. The "That's all, Folks!" card appears already pre-written and the firecracker explodes off-screen, rumbling and shaking the on-screen title card.
- When this cartoon aired on The WB, the part where baby Elmer points his toy gun at baby Bugs' face and baby Bugs cracks his bottle of carrot juice over baby Elmer's head was cut.
- See "Notes" section below for edits done to the end gag where the cartoon ends with elderly Bugs giving elderly Elmer a stick of dynamite after burying him in his own grave and the title card shakes in response to the off-screen explosion.
- In this short, Bugs Bunny in his normal adult age is not shown, just as a baby bunny and as an old rabbit.
- This is the first cartoon where something or something referring to the beginning happens at the end where the usual That's All Folks! ending card is shown.
- When an old Elmer is reading the newspaper, Bing Crosby's and Carl Stalling's names can be seen.
- The caption under Baby Elmer's picture reads "Only 3½ Years Old," a reference to Billy Gray's catchphrase.
- While Baby Bugs is babbling to Elmer as a baby, the words of his catchphrase, What's Up Doc, appears and Elmer reads it.
- A similar gag would show up in the later Popeye the Sailor cartoon Popeye the Ace Of Space. when Popeye is captured by aliens, they babble the words',... on this typical Earthman specimen, appears and Popeye reads it.
- Final cartoon to bear only WARNER BROS.. All cartoons after this will bear WARNER BROS. PICTURES INC..
- Final cartoon to use Produced by WARNER BROS. CARTOONS on opening titles.
- This cartoon would be used as the final cartoon for the documentary, Bugs Bunny: Superstar.
- This is one of the several Looney Tunes/Merrie Melodies shorts to have ending gags involving the closing titles. Others include: Porky's Duck Hunt, Stop! Look! And Hasten!, and Box Office Bunny (all of which have been edited on television in one form or another, due to some channels not allowing the cartoons to have their original ending cards).
- The USA Turner dubbed version which used to air on Cartoon Network and currently airs on the Latin American, Canadian, and American Boomerang (excluding an appearance in The Bob Clampett Show and the first "June Bugs" marathon) channels has the original shaking end card replaced by the 1947-48 Merrie Melodies card which doesn't shake (though the explosion sound is still heard).
- While the USA Turner dubbed version replaces the ending title card with the 1947-48 MM dubbed card seen on most USA dubbed versions, the European Turner dubbed version which airs on the Cartoon Network/Boomerang channels in European countries. preserves the original ending card and the explosion gag. In this version the original end card shakes, and the "DUBBED VERSION (C) 1995 TURNER ENTERTAINMENT, CO." disclaimer fades up at the end.
- A previously unreleased restored print of this cartoon with the original shaking end card aired on MeTV on 6 February 2021.
- The copyright was renewed in 1972.