Some of us don't equate songwriting with telling stories unless it's in the form of a folk song. Folk songs often have what is called an AAA structure, with "A" representing the verse without chorus, therefore lending themselves easily to a story format. Many of the 60's folk artists wrote in this style, which is in fact one of the oldest songwriting formats around.
A lot of pop songs on the charts these days don't rely very heavily on storytelling or even good lyric writing! I'm sure you can think of popular songs where you have no idea what it's really about, but you can relate to parts of it which has a great musical hook, so it's on your list of favourites. Does that mean you can get away with writing nonsense lyrics? Think again!
As far as stories go, any good one will draw you in from the first sentence, hold your attention all the way through and satisfy you with a great ending...maybe even leave you wanting more! When you write a song, even if it isn't in the folk or storytelling format, you need to remember to have the same elements ; a beginning, a middle and an end.
But what if there isn't a "story" in your song? How do you accomplish this task?
When you sit down to write lyrics, start with the first line before even knowing what the song is going to be about. As you add each line, an idea starts to unfold, and once you have a first draft you can more or less know what the song is about. Don't worry, most songs today don't even have any semblance of a story, so there is hope for you too!