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.
The cartoon starts with Elmer sitting under a tree, crying over never being able to catch Bugs. The "voice of God" (also the voice of Mel Blanc) 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 Anno Domini.
This offers the chance to use some contemporary gags with a futuristic twist, as Elmer finds a year 2000 newspaper. One headline says, Smell-O-Vision 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 cartoon ends with the "That's all, Folks!"(which appears every single cartoon) card appears already pre-written and the firecracker explodes off-screen, rumbling and shaking the on-screen title card.
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.
Also, instead of saying says it says sez.
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 have WARNER BROS. Present. All cartoons after this will have WARNER BROS. PICTURES INC. Present
Final cartoon to use produced by WARNER BROS. CARTOONS on opening titles.
On Cartoon Network (except for the first "June Bugs" marathon and The Bob Clampett Show), the shaking ending card is replaced with the generic "Dubbed Version" end card (though the explosion can still be heard), thus ruining the ending gag.
The European dubbed version which airs in EU countries keeps the original shaking ending card.
The WB! airing of this cartoon cut the part where baby Elmer points his toy gun at baby Bugs and baby Bugs breaks his bottle of carrot juice over baby Elmer's head.