If you're in for a learning experience, you could likely just spin up a regular old Linux VPS with an Apache or nginx server. Set up a vhost that just acts as a directory listing for some folder on the server. Since you're targeting old web browsers and just letting people download files, you could get away with not setting up SSL which would increase compatibility.

FTP would work too, since I think most web browsers can browse FTP as well - can't they?

I wouldn't rely on major cloud storage services, just simply because of SSL. If you've ever tried browsing the modern Internet on a Windows XP system with IE 6, you'd know exactly why. You're lucky if you can even get to a page that lets you download a more up-to-date browser without resoirting to using a more modern computer.