Hurricane by Bob Dylan
Bungalow Bill by The Beatles
Dinah Moe Humm by Frank Zappa
Meatloaf’s Paradise By The Dashboard Lights
Aw, come on. Let’s go ahead and mention one. 
El Paso, by Marty Robbins = best of the cowboy story songs
A Man Named Sue is the absolute greatest!
But my personal favorite right now is by the local band The Flametrick Subs. It’s entittled The Tiajuana Cat Toss. The joyful story of a frat boy in Mexico with twenty dollars to spare.
Stan by Eminem
Luka by Suzanne Vega
Battle of New Orleans by Jimmy Driftwood
“Well they ran through the briars and they ran through the brambles,
and they ran through the bushes where a rabbit wouldn’t go.
They ran so fast that the hounds couldn’t catch em,
on down the Mississippi to the Gulf of Mexico.”
But of course! 
The best story song by Weird Al is “Albuquerque”. A long, meandering story, but worth it! 
“A-L-B-U…QUERQUE”
Night of the Johnstown Flood by Bruce Springsteen
Percy’s Song by Bob Dylan
I guess he’s forgotten.
In his prime Tom T Hall’s band was called “The Storytellers.”
One of his songs…“Old Dogs, Little Children and Watermelon Wine.”
“The Wreck of Old 97”, many versions recorded. I like Hank Snow’s.
“Big Iron” by Marty Robbins
“Detroit City”, “Streets of Baltimore”, and “Lynching Party”, all by Bobby Bare
“Partners” by Jim Reeves
“Saginaw, Michigan” and “Long Black Veil” by Lefty Frizzell
That’s A Boy Named Sue.
AAH! 70s OVERLOAD!
My god, man, have you no saccharin receptors? That song is a Glurge Gallery in and of itself!
My favorite?
"Lemme tell y’all a story 'bout a man named Jed.
Poor mountaineer barely kept his family fed… "
(Actually, I’m with The Devil Went Down to Georgia by the Charlie Daniels Band.)
When the Beatles Hit America by John Wesley Harding, about a hypothetical Beatles reunion. He’s changed the lyrics a bit over the years, with references to DVD and pay-per-view, and I’m not sure if he still does it since the death of George Harrison (or Linda McCartney for that matter, she’s in the song as well.)
*And things skip into action immediately,
George buys the film rights in a flurry of activity.
Paul says they’re not gonna play Japan,
And Ringo cancels his chat show plans.
And John, who was never the quiet one,
John stays silent for time being.
And the world sings;
When the Beatles hit America,
When the Beatles hit America,
*[spoken]They’re gonna be sponsored by Visa.[/spoken]
Isaac’s Lament, by Uncle Bonsai, about a man who is distraught and flings himself from his apartment window when he hears news that The Love Boat has been cancelled.
There’s also the Doug songs, started with Uncle Bonsai and continued with various side projects, which could be considered one epic saga.
And, of course, A Dying Cub Fan’s Last Request by Steve Goodman.
There’s a beautiful English traditional (which I’m reminded of because it’s on the new Nickel Creak) called “House Carpenter”. Very lovely. Very depressing. My mother used to sing it to me.
That and “Detachable Penis”.
Um. I mean that’s another good story song. My mother never sang me “Detachable Penis”. 
Some good ones have been mentioned here, but my favorite has got to be this one: Greg Keeler’sLament of the Laundromat.
I heard it sung a cappella, in a plaintive Irish brogue. It never got much radio airplay–I heard it only once, but it has remained stuck in my heart like a switchblade. If anyone can hear this song without weeping, well, that person is not fully human.
Jumpin’ Jesus on a pogo stick, how could I forget:
Stuart by Dead Milkmen.
“They’re building landing strips for gay Martians, I swear to God.”
MrO, for a happier take on a similar theme, try Share a Load, by The Bobs (part of The Laundry Cycle).
That song would be Hazard, indeed by Richard Marx. Was released in 1992. This I know, because it appears on a compilation I have dating from my younger days entitled The Best of '92. Unfortunately, it is the worst case of false advertising since the film The Neverending Story.
So mny mentioned already, but the first one i thought of was Al Stewart’s “Roads to Moscow”.
Why has nobody mentioned the classics?
One Tin Soldier by Coven
“…Peace on Earth was all it said”
Billy Don’t Be a Hero by Bo Donaldson and the Heywoods
“… I heard his fiancee got a letter that told how Billy died
that day”
Run Joey Run by David Geddes
“… I ran to her. I held her close. When I looked down
my hands were red”
I apologize for the ommission of the spoiler warning
How about The Ballad of Curtis Leow by Lynyrd Skynyrd?
Ringo.
Or its companion piece, Irving, the 142nd Fastest Gun in the West.
No idea who sang either.
Regards,
Shodan
Although Big Bad John is almost equally classic.