Yes this certainly can be done.
You need a script to handle the upload/download for you and present it in a pleasant manner.
I can point you to a free script that is fairly nice.
I did not write the script.
I do not know the person that wrote the script.
I cannot provide support for this script. (In other words please do not ask me to log into your server and set it up for you) Other than that I will help as much as I can. (Also if something breaks on the script- Like I said I did not write it) please do not complain to me.
I did read over the script and it seems fairly nice and straightforward for installation.
The script is written in php (so your host needs to support php)
According to this script your host must also support .htaccess (user login/password)
The link to the site is below. The file I am referring to is called DODOUPLOAD Ver 1.2
Code:
http://www.regretless.com/scripts/
P.S. And before I get flamed for offering a script. I presumed he wants to have something nicer looking than simply uploading to a public directory on his server and letting the server display the directory tree. (you could do that instead) by using and ftp program like cuteftp or wsftp or winsftp3. Any of those programs will let you upload to a directory on your server. If you then navigate to that directory using a web browser. The browser will display the page like a directory tree.
Good luck on your decision.