"Dropbox"

Ross, I don't think something like that exists. But it's not actually hard to make, really.

• Apache can be configured to redirect any PUT request to a given script. Just write yourself a small PHP or CGI script that will take a file, put it somewhere safe, and send a 200(ish) status code back. PHP has a function for precisely that purpose.
• Writing something to give you an HTML form back if someone does a GET / isn't exactly hard.

Altogether, it sounds like a bit of tinkering that'll be finished in an hour or so.