Railroad Earth songs all tell a story. There’s one about a hunter who finally gets his elk/ deer and takes a bite of the warm heart.
Not sure if these qualify:
“Piano Man” - Billy Joel
“Tom’s Diner” - Suzanne Vega
“Me and Bobby McGee” - Kris Kristofferson or Janis Joplin
“One Bourbon, One Scotch, One Beer” - George Thorogood
“Laurie (Strange Things Happen)” - Dickey Lee
“Patches” - Dickey Lee
How could you leave out,
Well, Earl grabbed on the shifter
And he stabbed her into fifth gear
And then the chromium plated
Fully illuminated
Genuine accessory shift knob
Come right off in his hand
I says, “You wanna screw that thing back on, Earl?”
So far as I can tell, nobody has mentioned Tangled Up In Blue. Must be a conspiracy - so I won’t either.
However, let me introduce you to (or remind you of) Dat by Pluto Shervington (1976). It’s… not an easy story to follow, as the details are recounted by Pluto in Jamaican patois - but it is splendid. Wiki describes it as:
…about a Rastafarian [Ozzy] trying to buy pork (without naming it aloud), contrary to his faith, so that he can afford marijuana
(This went unbanned in the UK at the time, presumably because none of the worthy souls at the BBC could understand it.)
In the course of his efforts, the butcher tries to dissuade Ozzy, leading him through possible alternatives to pork, at one point mentioning the meat by name, leading Ozzy to exclaim:
Hush your mouth man, me brethren hear
Sell I a poun’ a dat thing there
Having obtained his pork (and so ensured he has enough money for weed), Ozzy then runs into Rasta Jeremiah on his way home, leading to an unexpected conclusion to the story.
I wrote that explanation 'cos I’m kinda evangelical about this song, and I’m trying to persuade you to give it a go.
j
Here’s a couple I never thought I’d mention in the same breath:
The Gift (Velvet Underground)
Detachable Penis (King Missile)
Red Barchetta, by Rush
Oh, I didn’t see this in the original post. That knocks out all the songs I posted above.
Here’s one that’s part spoken, part sung, Bette Middler’s “Otto Titsling”. It isn’t structured with a chorus, though.
Haven’t thought of this one for years, but I used to love it as much as Alice’s way back when.
Jaime Brockett - Legend of the USS Titanic
If you like that, also try “Talkin Green Beret Super Yellow Hydraulic Banana Teeny Bopper Blues” off the same amazing album, Remember the Wind and Rain, which, as one would expect from the title, contains regular folk songs as well.
My first thought was 30,000 pounds of bananas, but that’s already been mentioned. This song then occurred to me as I went through this delightful thread.

I found out about this one from the Dope!
I sang “The Mariner’s Revenge Song” to my mother when driving from Arkansas to California. I think we were in New Mexico so it’s not like it took me the whole trip to sing it. It’s not that long.
Ghoultown’s “Drink with the Living Dead” tells the story of a cowboy who gets into a drinking contest with dead guy cursed to walk the earth.
‘Jailbreak’ by AC/DC. It’s about… well, I don’t want to spoil it.

“One Bourbon, One Scotch, One Beer” - George Thorogood
John Lee Hooker did it better ten years before Thorogood. And Amos Milburn originally recorded the song in 1953, over ten years before Hooker.
Trivia: the songwriter originally penned it One Scotch, One Bourbon, and One Beer.