How can I create a file (let's say a markdown file) that persists on the server, strictly using HTML5? So no database or web server, just an static site server.
Your question is probably too broad but you need some type of program that serves a file but not "server software" such as Apache or nginx. Just like those web servers, they are written to listen to a port on the IP address of the computer attached to an internet connection somewhere. A program to do that is actually relatively easy to write but all the needs of a proper web server aren't such as needed security and the ability to interface to other programs.
But then that program is, in essence, a server and functions in the same manner so being able to serve a file without a server of some sort in some manner isn't possible.
Whole books are written about this and one of the best is, "Advanced Programming in the Unix Environment" by Stevens and "Unix Network Programming" also by Stevens.