If I understand correctly, you want a self hosted version of Wattpad or Inkitt for posting stories. I don't know if there is a 1:1 software that can do exactly that.
As a starting point, you could try building a static website. They easiest ones follow a blog format and store the text on a markdown file. Many have recommended Hugo on this thread, but I'd like to recommend Zola instead. Paired with the well documented tabi theme, you could spawn a new "blog" for each story and use the archive view to have the chapter selection.
It's not exactly the best solution, but I think it's a good starting point to try it out.
P.S. Zola is inspired in Hugo. The creator thought Hugo was too complicated.