I have a problem in understanding this type of relation between object and possessive pronouns. Can someone help me with this?
"The story of ours", or more likely "a story of ours", most likely would mean a story that belongs to us, in the sense that we own it; maybe we own the copyright. It doesn't mean that the story is about us.
"The story of 'us'" means a story about us, that is, we are the subject of the story. Note the extra punctuation that I put around "us". This is to show that it is an unusual use of the word, meaning "us as a couple", as pointed out in the comment.
Now, if it said "our story", that could have either meaning: owned by us, or about us.