Online backup, legal issues

Let us take an example, dropbox. Quotes are from the terms page. 

Your Public Folder

While you own the content contained in Your Files, files placed in your public folders are automatically available to other Dropbox users and to the general public. By placing Your Files in your public folder, you hereby grant all other Dropbox users and the public a non-exclusive, non-commercial, worldwide, royalty-free, sublicensable, perpetual and irrevocable right and license to use and exploit Your Files in your public folder. In other words, a file in your public folder can be used by anyone, for any purpose except commercial use. If you do not want other people to be able to use Your Files in this manner, then simply do not place Your Files in your public folder. By placing Your Files in your public folder, you agree and acknowledge that Dropbox has no responsibility or obligation to monitor or notify of you of any non-compliance related to the license you have granted and that Dropbox has no responsibility to enforce or police, or aid you in enforcing or policing, the terms of that license.

A few paragraphs later, we see that dropbox complies with DMCA.