Apple - Can OS X access files in my Dropbox account without using space on my local hard drive?

Update: See nassimhddd's answer below about ExpanDrive allowing you to mount your DropBox account as (external) drive - since its Mac version 3.0.3021 May 2013.

The short answer to this is: No. As far as I'm aware, you can't mount your Dropbox account as some sort of remote file share - its synced files are always written to your local disk.

That said, you can change the location of the Dropbox folder away from its default of ~/Dropbox, so if you have a separate internal or external disk you could put the Dropbox folder there.

You may encounter problems if Dropbox occasionally can't access the detached volume holding its folder, however. See here:

Your Dropbox folder must be available when your computer boots and before the Dropbox desktop application starts. Some USB drives, for example, take some time to mount before your computer recognizes them. The Dropbox desktop application could start before such a drive is ready and get confused. You will receive an error and Dropbox will give you the opportunity to relink your Dropbox account or exit if it is unable to find your Dropbox folder.

ExpanDrive

Since its Mac version 3.0.3021, ExpanDrive makes this possible. I just tested it and am very satisfied: very easy to setup and the folder navigation is very reactive (which was not the case with DropDAV). Plus it works with many other services (google drive, box, etc.).

You can use a third party service like Otixo or DropDAV that will allow you to connect through webdav protocol. Unfortunately I don't know anything free.

Anyway after you have create your account you have just to press cmd+k in the finder and connect to the server using the new credentials. This will mount a virtual drive with Dropbox content. You can also mount the drive at startup adding it into login items list.