The only rule was that there had to be a fairly coherent narrative that was continued from verse to verse. And I left off show tunes since the narrative isn't entirely contained within the song.
1. Johnny Cash, The Streets of Laredo
A traditional song, but it's entirely his - especially the cover on American IV. Cash changes the lyrics a little - the cowboy has been shot instead of cut down by a venereal disease - but the original lyrics would have spoiled the mood, as venereal diseases so often do. Cash's cover of Long Black Veil might also belong on this list, despite the fact that he starts laughing in the middle (another mood spoiler).
2. The Wrens, 13 Months in 6 Minutes
This song has one of the best sets of lyrics I've ever seen - even if you have to read the liner notes to follow them at all, since they are mumbled and low in the mix.
3. Joni Mitchell, Green
4. Bruce Springsteen, The River
Another unbearably sad story. Story songs tend to be either incredibly sad or funny. As a corrective that doesn't merit being on this list, here is Ray Stevens singing I Am My Own Grandpa.
5. Loretta Lynn, Van Lear Rose
A love story about her own parents. Country music seems to be the only genre (with the possible exception of rap) that doesn't seem to have given up on the story song.
6. Outkast, Da Art of Storytellin' (Part I)
Actually two stories back to back - its place on the list is largely merited by Andre 3000s contribution (the second verse).
7. Bob Dylan, Clothes Line Saga
I realize that Dylan has probably written "better" story songs, but I can rarely listen to Hurricane or even 115th Dream without wishing them a little shorter (Lily + whatever + Jack of Hearts I skip entirely) - this song I always listen to, and I can't imagine anyone else even trying to imitate it, let alone write it.
8. Townes van Zandt, Pancho and Lefty
His version at the Old Quarter is far superior to every cover I've heard.
9. John Prine, 6 o'clock News
Prine is another great narrative songwriter. The song follows a boy from birth to death ("Wanda had a baby, in 1951 / the father was a stranger, a stranger was the son") - with each repetition of the chorus going back to the moment of the boy's conception ("C'mon baby, spend the night with me").
10. Ron Sexsmith, Strawberry Blonde
Another tale of childhood, happier this time -- seeing a girl from elementary school years later grown up with her own child. The element of time can lend story songs a special poignancy that can't quite be duplicated by non-narrative lyrics, I think.
11. Gillian Welch, One More Dollar
A tale that everyone can relate to - an early frost prevents us from getting work picking fruit trees, and so we turn unwisely to gambling to try to pay our way back home.
12. Richard Thompson, Beeswing
A gorgeous gorgeous song. It's pretty amazing how long he's been good.
13. Guy Clark, Let Him Roll
A little sappy, but everyone needs that sometimes. "It was white port wine that put that look in his eyes, that grown men get when they need to cry..."
14. Pulp, Disco 2000
It is dangerous to attach a year to a song, but for some reason it works on me even more now, since I have to think about how long ago I listened to it. Won't it be strange when we're all fully grown?