If an author is not concerned about making money from her story, but rather would like to publish the story to protect the story from copy right yet allow others to read and review the story for critical improvement, is there such an open source website or facility that promotes such a peer review environment?
Create an ebook and distribute it through Amazon's Kindle Direct Publishing (or other ebook selling platforms) for $0.
Amazon KDP allows to give away your book for free for some limited time, after that you have to sell it for at least $0.99. But you can publish it for free elsewhere, and since Amazon always wants to have the cheapes prices, once you report that your book is available for free elsewhere, you can lower your price to free on Amazon, too. I never tried this, just read about it in forums and blogs. Google to find a more detailed how to. Of course you can always ignore Amazon and just use other platforms, but Amazon has the widest reach, so I would try to be on there, if I could.
Don't be hasty and throw away your idea just yet. Take some time to evaluate all options, let some experienced readers read it and give you feedback (find bloggers doing reviews in your genre and ask them for feedback), if they are even half positive, either educate yourself to write well or find a co-author, etc. You can always publish your book free, but maybe some better ideas come along in the next weeks or months.