Most of the ones that I would call great have been named earlier, but here are a couple that I remember listening to during my S.E. Asian vacation, way back in 1969.
Trouble Every Day, F. Zappa
Ballad of the Hulk, Jerry Jeff Walker
Although Red Barchetta is excellent, I think The Trees is a bit better.
And one that I think is a bit obscure is The Outsider which comes from one of the greatest yet most overlooked albums of all time, You’re Never Alone With a Schizophrenic by Ian Hunter.
Oh, and since nobody mentioned it yet, Bohemian Rhapsody.
I don’t think this one has been mentioned yet…
The Scotsman by ?
About the Scottish bloke who falls asleep under a tree and a couple of young women wander by and take a peek beneath his kilt. When he wakes up there’s a blue ribbon tied around his…uhhh…willy. Hilarious!
I have to second all the Kenny Rogers songs, plus name El Paso by Marty Robbins.
But the most fun one is Watching Scotty Die, by the Dead Milkmen. 
I can’t believe I forgot Too High for the Supermarket by the Uninvited.
Also, Gin N’ Juice especially as covered by Phish.
Fairytale of New York, by Kirsty MacColl* and The Pogues
Her version of Miss Otis Regrets is good, too (echoing an earlier vote).
Two Hot Girls on a Hot Summer Night, by Carly Simon
And at the possible risk of being accused of schlock-mongering, All I Wanna Do Is Make Love to You, by Heart. In part because I didn’t “get” the final lyrics on the first listen. (Don’t laugh; neither did Ann Wilson!)
“Comin’ After Jinny” by Tex Ritter.
Written by that ole cowboy himself, “famed” country song writer Shel Silverstein.
Oops! And one of his other songs is also in this list!
Another vote for “Mr. Tanner” from Harry Chapin. Also “Suzanne” by Leonard Cohen, and a bunch from another great story teller, Jimmy Buffett…“Spider John”, “West Nashville Grand Ballroom Gown”, “A Pirate Looks at 40”, “He Went to Paris” (okay, i really like his early stuff best :). And above all of the Buffett songs… “God’s Own Drunk”!
“The Scotsman” is by Mike Cross
Bryan Bowers does a version also.
How about The Road Goes on Forever sung by the The Highwaymen (Willie Nelson, Waylon Jennings, Johnny Cash and Kris Kristofferson) and written by James Earl Keen Jr.
Already named, but I want to add my votes for…
Taxi - Harry Chapin
Paradise By The Dashboard Lights - Meatloaf
Sittin’ there on that sack of seeds…
Sink the Bismark by Johnny Horton
Leader of the Pack by
Can’t give you Irving…, but Ringo* was brought to the airwaves by Lorne “Ben Cartwright/Alpo Man/Voice of Doom/Battlestar-Galactica-guy”* Greene.
*[sub]Not necessarily in the order given[/sub]
Rocky Raccoon, by the Beatles
Escape (The Pina Colada Song) – not really, but I had to mention it anyway!!
The Unicorns, by Shel Silverstein (sung by The Clancy Bros & Tommy Makem)
Wavy, a post like that could get you hurt where I’m from! :mad:
First off, it’s Robert Earl Keen. (and James Earl Jones was Darth Vader) 
Second, though I haven’t heard the Highwaymen version, anything other than Robert Earl’s own performance of The Road Goes On Forever just wouldn’t be right.
Since you mentioned The Highwaymen, the song The Highwayman is a good story song.
Just wanted to clear that up. We take REK seriously around here.
Okay, for a seriously sick but still kind of fun story song - “Country Death Song” by The Violent Femmes.
<lyric sample>
I gave her a push, I gave her a shove.
I pushed with all my might, I pushed with all my love.
I threw my child into a bottomless pit.
She was screaming as she fell, but I never heard her hit.
She was screaming as she fell, but I never heard her hit.
</lyric sample>
Lyrics here
“The Ballad of Irving” - Frank Gallop
Kalashnikov has already mentioned Copperhead Road by Steve Earle, who for my money is one of the best “story” songwriters working today. Two Earle favorites of mine are Tom Ames’ Prayer and Taneytown.