Most of the songs that I play tell a story.
I don't know the style of music you like, but here are a few suggestions that I like to play:
Most of the songs that Harry Chapin did were stories. Taxi; W.O.L.D.; A better place to be; 30,000 pounds of bananas, to name a few.
Kris Kristoferson sings some good stories, To beat the Devil; Sunday mornin' comin' down; Bobby Mcgee; Best of all possable worlds, lot more.
John Denver sang a lot of stories; My sweet lady; It's good to be back home again; Poems, prayers and promises, etc.
Jim Croce sang a lot of stories.
Operator; I'll have to say I love you in a song; New York's not my home; Workin' at the car wash blues; Rapid Roy(that stock car boy) many more.
If you like funny songs, Pinkerd and Bouden (hope I spelled Bouden right) do a lot of them. I don't know if you have heard of them or not, but, they are a hoot. Help me make it through the yard; Don't pet the dog; Blue hairs drivin' in my lane, etc.
The Unicorn by the Irish rovers is kind of a fun story.
Most of these songs are kind of old, but the people that I play for seem to like them.
Like someone else mentioned, probably about 40% of songs tell stories.
Hope that helps you some.