Actors drool over characters like Joe Exotic and Carole Baskin. They’re big in very specific ways. They’re obsessive and damaged, and they hate each other. He’s a colorful sociopath. She’s awkward and painfully aware of it.

John Cameron Mitchelland Kate McKinnon own “Joe vs. Carole” from the start. They turn middling and well-worn material into a deeply engaging human (and feline) drama. They take an outrageous story — lions, tigers, bigamy, murder for hire — and play it straight down the middle, knowing there’s little need to exaggerate the absurd. There’s no mockery here. It would be redundant.

You may already know the story of Joe and Carole from the Netflix pandemic-era hit true crime series “Tiger King,” or the Wondery podcast “Joe Exotic: The Tiger King,” on which this new Peacock limited series is based. Exotic ran an animal park in Oklahoma, where he featured big cats and didn’t take very good care of them. Baskin runs a nonprofit animal sanctuary near Tampa, Fla., where she wages war against those who hold big cats in captivity and use them for profit. Joe fits the description in spades. She went after him in the courts of law and public opinion. He responded with a string of profane videos in which he suggested, among other things, that she killed her second husband.

Then he put out a hit on her life.

But let’s talk some more about Mitchell, who came out as gay during a visit to San Francisco in 1985 and went on to create the cult stage musical and movie “Hedwig and the Angry Inch.” He does something tricky here. He infuses a lunatic with something resembling dignity. Strutting and shouting across the screen in a bleach-blond mullet, handlebar mustache and mascara, Mitchell embraces Joe’s redneck side and his loud, proud gayness. We hate him and we love him; we feel for him and, eventually, fear him, just as Carole does. He’s like a psychotic rooster.

Meanwhile, McKinnon, the longtime “Saturday Night Live” standout, relaxes into the role of Carole. With arms stiff by her side, and a penchant for swallowing her own sentences before they’re fully formed, McKinnon fully fleshes out a character rarely at ease with herself but confident she’s on the right side. Carole has had a rough life marked by physical and emotional abuse; when she finally meets a kind, supportive man, Howard (Kyle MacLachlan, bringing understated quirk to a nice guy), she’s not sure if she deserves him. Carole is a smart variation on the crazy cat lady, except her cats are a tad larger than most.

Created by Etan Frankel, “Joe vs. Carole” is the tale of a needy huckster and the dead-serious woman who becomes his white whale. There may be a sucker born every minute, but Carole refuses to play the part. She’s a lot easier to like than her adversary, but it’s Joe — and Mitchell — who steals the spotlight. Just when you think TV has nowhere to go with the antihero, along comes this American original. He’s hard to like, but good luck averting your eyes.

M“Joe vs. Carole”:Limited series. Starring John Cameron Mitchell, Kate McKinnon, Kyle MacLachlan, Nat Wolff, Sam Keeley, William Fichtner, Marlo Kelly, Brian Van Holt and Dean Winters. Created by Etan Frankel. (TV-MA. Eight episodes at approximately 60 minutes each). Available to stream on Peacock starting Thursday, March 3.