Question in title. Is there any way for such configuration? OR at least that files are stored only on one computer. Of course I am aware that I can use internet browser version of dropbox, but I would like to use the desktop one, but exactly in the way a browser version works, i.e. without losing hard drive space.
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.
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.
Here is the solution. Dropbox is great! Selective syncing. https://www.dropbox.com/guide/business/manage-files/save-your-hard-drive-space
This is a little bit old question, but yes, this can be done very easily.
ExpanDrive is an app that does that. It works not only for Dropbox but also for Google Drive, S3, SFTP WebDAV and many other cloud storage solutions.