Is it possible to actually publish my fanfiction?

I recently finished a naruto fanfiction and have recevied extremely positive feedback and a decent amount of people have told me that I should really consider changing the names of the characters and publishing it. To summarze it simply (it takes place in our world) the story is about a 21 year old man (sasuke) who for some reason is immortal, finally finds the reincarnation of his past love (hinata) after nearly 500 years of searching. His joy is short lived however, for he soon realizes that she is happily married (to his boss, Gaara) and has a small child. All the same, Is he truly willing to give up on her? I guess what Im asking is, do you think a story like that could get a shot at being published, and could it be published in general? I have yet to finish uploading the whole story on fanfiction.net but if you want an idea, it's call Los Inmortales (the story is in english) under the rated m section (lemon scene) lol anyways yepp and for thoes who watch naruto srry about the crack pairings xD