The soul of a story? This depends entirely on what your story is. If it's a plot based story, then the soul would be the plot. If it's character driven story, then the characters would be the soul. It all depends on what the main focus of the story is.
And that's the really 'pretty' version of this. Call me jaded, but I don't believe there's a soul to a story. Right now, the book I'm writing is filled with things that I just wanted to read about. A stressed out skeleton, his human wife, their illegal work, and the world they live in. There's a theme, but only because I wanted to answer a question in a messed up way, that ANOTHER piece of work was asking. There's no 'soul' in my story, just...It's literally just a story. It's not set to change the world, make people think, or barely entertain. It just kinda exists, and if it manages to do those things, well good on it.
Saying that a story has to have a soul is what sets a writer up for failure. Since writers naturally live in a constant state of self-doubt, they'll never be able to find that soul. It's best to let the readers find the so called soul, than go into writing a manuscript hoping to make one for the story.