It's always dangerous to disagree with SuKi, but "My one is a long story." rattles my US native ears. I'm not sure what "one" refers to here. My life? I'd always say "Mine is a long story" or "My story is a long one."

My one is a long story. or Mine is a long story.

I'd say "My one..." as a follow on to someone else's story. If Fred says HIS story is about jellyfish, I might say, "MY one is about goldfish." Obviously there the 'one' is 'story.' The way the student has set the question I am not sure if it is a dialogue. I think 'one' would be 'lifestory' (is that one word?). I would reword the question and then I think my/mine/my one could each be used in particular contexts.

You could say "Mine is a long story." OR "My story, is a long story." Between these it depends on the context. If someone asks you a question and knows that you are telling them about your life story, then you would use mine. However if it has not been stated earlier in the conversation, that you were talking about your life story, you would use "my story".