The title is a play on the expression "neither hide nor hair."
Bugs comes out of his rabbit hole in a city park every morning because a kind gentleman (who has the likeness and mannerism of Peter Lorre) keeps coming to feed him a carrot ("Well, here I go again with the 'timid little rabbit' routine. It's shameful, but- eh, it's a living!"). At first feigning a usual rabbit posture, Bugs eventually stands up and confides that he'd rather simply go home with the gentleman as a "pet", since it would be easier on both of them. As the gentleman brings Bugs home, he remarks that it is strange that Bugs calls him "Doc" because "I happen to be a doctor." The camera then pans up to see that the name above the apartment is none other than Dr. Jekyll.
Inside the house, Bugs acclimates to his new surroundings. Dr. Jekyll comes across a bubbling potion that he knows he shouldn't drink, but he relents ("Oh, I'm so ashamed!!") and drinks the potion anyway. He then transforms into the ogre-ish Mr. Hyde, with a monstrous green face and glowing red eyes. Bugs tries running from this monster, but soon the monster reverts to Dr. Jekyll. Bugs, thinking that the monster is still chasing both of them, tries leading the doctor into various rooms and closets, with the eventual retransformation of the doctor into Mr. Hyde. This continues for a few minutes, until Dr. Jekyll decides that he's going to pour the rest of the formula down the drain. He goes into his lab, but finds only the empty beaker. The Doctor asks Bugs if he drank the potion; Bugs becomes insulted at the idea and leaves ("I am going back to the park! There is no question of my integrity there", he huffs).
Walking back to his park, Bugs transforms into a monstrous green rabbit (an implication, to the viewers anyway, that he DID drink the formula). The people at the park who are busy feeding the pigeons see Bugs and run away screaming. Bugs looks at them and wonders, "Now what's eating THEM? Hmph! You'd think they never saw a rabbit before!" After Bugs Bunny says his last lines, he eats a carrot and watches the old ladies and old man run away as the screen closes.
In the cartoon, Bugs utters the phrase, "I wish my brother George was here" as he sits down to play a piano. The reference was to the famous pianist Liberace, who had a brother George, who was also his conductor on TV and personal appearances during the 1950s.