The "story" in "Ode on a Grecian Urn" is pretty basic as it hardly exists at all as the "characters" are pictures on an ancient Greek vase. So, really, if there is any story it is about the poet looking at the urn, thinking about it and the ideas it inspires and then writing these down. Of course, the picture represents a story. That story seems to contain two sweethearts. The male character appears to be interested in the female character and is trying to woo her. There are trees around with branches that are in leaf. There is a musician playing the pipes with songs that will be for ever new.This is because they are a snapshot in time and nothing in the picture story will ever age or become ugly. There seems to be a procession, as if it's some sort of celebration or ritual. All the characters are stuck in a time warp and Keats is playing with the idea of Time/Space/Motion/Gravity. He invents imagined places for the characters, but really, even if time never moved on for them, it does for everybody else including us.
In this poem, the narrator has found an old urn (like a vase) that was made long ago by some Greek person. The urn is decorated with two scenes -- one of lovers listening to musicians under some trees and one of a priest leading a heifer to be sacrificed on an altar.
So the narrator is looking at this urn and contemplating what it means.
First he thinks it would be great to be the lovers or the musicians because they never die. Their love and their music live forever.
But then he starts to think about how the lovers will never actually touch and make love, the musicians will never be heard by actual ears. And the town where the people are coming from to the sacrifice -- it will always be empty and desolate.
So by the end of the poem he's kind of conflicted and he thinks that because we're mortal we have to find something other than beauty to rely on.