There is a famous story that has been made into a film several times. If I wrote it in my own style, could I have it published as another version of the original?

I had a brief look into this (I'm by no means an expert), and there doesn't seem to be a straightforward answer.

As an example, I'd recommend checking out the Wikipedia page about legal issues with fan-fiction (https://en.wikipedia.org/wiki/Legal_issues_with_fan_fiction) because the cases mentioned all center around the issue of works that are "derivative" of copyrighted material.

Ultimately though, I don't think it's impossible for you to be successful with your endeavour... Just look at 50 Shades of Grey!