Create an "under construction" or "updating" page. Name the file index.html and upload it. This will replace your home page with the message of your choice while your site is down.
Because of the availability of on-line editors, you will find that very few people have local copies of their web pages. Many don't even know that they can have local copies, or how to deal with them.
Of course without a local copy as backup it only takes one minor glitch on the server and their site may be gone and any server backup may be quite old.
Also replacing the home page only disables the home page. Any decent site will have lots of internal links as well. You can't really disable a site once it is uploaded. That's why most professionals leave the old version live until the new version is finished and then replace the whole thing in one go.