Only if you’re a “Father-Raper”! 
I don’t know though, Dewey. That one’s kinda “on the cusp”, ya know?
Yeah, it tells a story, but it takes Arlo 15 plus minutes to tell it, and he “talks” a bunch of it.
I consider that one more of a comedy routine with musical accompaniment.
Now if you had menitoned Last Kiss by J. Frank Wilson and The Cavaliers, you would have shot right up to number one!
Q
I came in to mention Lily, Rosemary & the Jack of Hearts.
‘The Art Teacher’ by Rufus Wainwright on ‘Want Two’ - amazing (live) performance, great story.
Darn you! and you too, commasense!
And 'Hit Somebody! (The Hockey Song) by Warren Zevon
And of course ‘Boy Named Sue’
MiM
When my brother and I were kids, we would “play Hurricane”, which meant that we put on the record and acted out the song. Love it!
“The Blind Man in the Bleachers” by Dave Geddes (although I think the original name might have been “The Last Game of the Season”).
“Long Black Veil” is one of my favorite story songs. I am most familiar with the Johnny Cash version – it really gives me the shivers when I hear it, it is so sad and creepy at the same time.
Harry Chapin has a funny section of the live album where he discusses his attempts at finishing “30,000 Pounds of Bananas”. “Include something about a mother because the song already had a truck”
I started to wonder if the watchman made up the first story just to get the waitress to go home with him.
I like all of Harry Chapin’s stuff, particularly “Flowers Are Red”, “The Rock”, “Corey’s Coming”…creepy, moody sad stuff!
I’ll nominate the classic American Pie…
Jukebox Hero – Foriegner
Roland the Headless Thompson Gunner!!!
And I’m also partial to One Bourbon, One Scotch, and One Beer. (Any time anyone in my immediate circle tells someone else that they’re funny, the other person is sure to respond, “Everybody funny. Now you funny, too.” 
)
Travelin’ Soldier as done by the Dixie Chicks.
“The Biggest Ball of Twine in Minnesota.” Weird Al.
I love an old folk ballad, “The Flower of Northumberland”. A scottish knight is imprisoned by the Earl of Northumberland, and he persuades the earl’s daughter to free him by telling her:
It’s, “Oh, if a lassie would marry me,”
Oh, but her love it was easy won!
“I would make her a lady of high degree
If she’d loose me out of this prison so strong.”
She helps him escape, but:
As they were riding across the scots moor
He said “Oh, but your love it was easy won!
Get you down from my horse, you’re a brazen faced whore
And you need to go back to Northumberland.”
I like “Scenes From an Italian Restaurant” from Billy Joel.
When I’m naughty, or driving, I also dig “Family Snapshot” from Peter Gabriel and “Three Days” from Jane’s Addiction.
Ack…forgot to include One Tin Soldier
Reminds me of Stan Rogers’ “Oh No, Not I”:
A Newfoundland sailor went walking on the strand
He spied a pretty, fair young maid and took her by the hand
“Oh will you go to Newfoundland along with me?” he cried
But the answer that she gave to him was “Oh no, not I.”
“If I were to marry you, on me 'twould be the blame
Your friends and relations would scorn me to shame
If you were born of noble blood and me of low degree
Do you think that I would marry you? It’s Oh no, not me.”
<snip clever lyrics>
Eight months being over and nine coming on
This pretty fair young maiden she brought forth a son
She wrote a letter to her love to come most speedily
But the answer that he gave to her was “Oh no, not me.”
<snip more clever lyrics>
“The Road Goes On Forever” by Robert Earl Keen–which I first heard performed by the great Joe Ely.
Townes van Zandt’s “Tecumseh Valley” is another fine one. Has anybody mentioned Peter LaFarge’s “Ballad of Ira Hayes”? This is one of the “covers” Townes sang–but Johnny Cash’s version is well known.
The Green Fields of America’s version of “Reynardine” on their Live album is a haunting version of an old tune. And their “Kilkelly”–compiled from Irish letters to relatives in The New World–has been known to bring tears.