I presume that (a) it's possible and (b) it's been done before, but I can't seem to quite get it right. I followed the Install Dropbox via command line instructions at https://www.dropbox.com/install?os=lnx but I always get the error:
Couldn't start Dropbox This is usually because of a permissions error. Errors can also be caused by your home folder being stored on a network share. Get more help at https://www.dropbox.com/c/help/permissions_error?cl=en_US
The problem with permissions with Plesk systems is that the user folders are not writeable by the (domain) user (yeah, you read that right), but the httpdocs is. Interestingly, when I ran this as root, just to test, it worked fine. It created the .dropbox file and Dropbox folder and the files started syncing from my account. I thought I'd get away with chowning those to the domain user that would be using dropbox but I still get the exact same error. As far as I can make out it's not using any other files.
As another test, I set up a test domain and chown'd everything in there from root:psaserv to domainuser:psacln, but again it wouldn't work like that either.
I'm on Centos 5.6 with Plesk 10.4