Of course it's possible to provide a service which only receives and stores encrypted data. At that point Dropbox would cease to exist. Half the posts about dropbox are lauding it for doing one simple thing well. Your Dropbox/ folder is shared with no fuss.

The other half of posts about dropbox are ribbing on it for not being secure enough.

Here's a clue: If I had to carry an private key with me to decrypt data stored in dropbox i might as well just carry the data.