I have absolutely no idea how to do this using Perl or any kind of CGI.pm trick, but I would do this using symlinks on the server filesystem:

Use the docroot as a kind of holdall with index.html or index.cgi directing them to a page in a subdirectory. The place symlinks in the docroot to all of the desired files.

I am NOT suggesting you do this, though: as a solution it is highly crufty. In spite of making a mess there are probably other good reasons not to do it of which I am unaware. It was just something that occurred to me.

Elgon

Update: One thing I forgot: Why? (See also Salvadors reply about mapping below.)

Update #2:Okay, now I understand the kind of question we're dealing with here. What I would do is use a script which basically takes your account details and uses some kind of ID authentication function (password, credit card details, account details etc...) to take an encrypted file from a directory visible to the webserver, unencrypt it and return it to the client: then it doesn't matter if the URL is visible 'cause no bugger can unencrypt it without the key. Of course they're welcome to try a brute force attack if they want...

On the other hand, if they really want to hand off the data to someone else, they can just take the copy they have, copy it and email it to them. Never underestimate a resourceful idiot!