Thanks for the suggestions guys, I’ll have to look some of these up.
I endorse this post. Kevin Smith should be famous for his stories even if he never wrote or directed, and Chris Titus has the whole tragicomedy thing down pat. Both great choices.
Even though the OP specifically asked about comedians who are great storytellers, others have brought up raconteurs and songwriters. So I thought I’d mention a couple of other things that sprang to mind.
Anyone who likes live storytelling should check out The Moth podcast. Not all of the stories are comedic, and some are rather dark, but there’s some good stand-up storytelling there.
If you’re into that sort of thing, some preachers are great storytellers and include funny and/or touching stories in their sermons. One example of a great preacher-storyteller is Tony Campolo, who has some sermons available free online. This is, of course, evangelistic Christian preaching, so don’t listen if it bothers you to be called to follow Jesus (but it’s good evangelistic Christian preaching).
David Sedaris. David Sedaris!
I’ve never read one of his books but I’ve listened to 3 of them on audiobook. Hearing him tell his stories can’t possibly be beaten by reading his stories.
You can get a few of his readings for free on podcast from NPR I believe.
An old-timey story teller you might enjoy is Jerry Clower. Very southern cornpone, but his bits are usually very funny. Example.
How 'bout Brother Dave Garner he is the master of the hairy dog story…