Most commonly, it's a noun:
* Example: "I read a fascinating story about a talking cat."
It can also be a verb:
* Example: "She storied her childhood adventures to the children." (meaning she told stories about her childhood)
In very rare cases, it can even be an adjective:
* Example: "The storybook illustrations were beautiful." (though "storybook" is more commonly used as a single adjective)
So, the answer depends on the context. Without a sentence, "story" is most likely a noun.