The answer varies by moodle version.
In my v2.4 installation, within the side-menu /settings / assignment administration / download all submissions is an option. It bundles all the files submitted into a zipfile for you.
However, I do not find any way to download all the OnlineText or SubmissionComments. That is one reason I now learn more toward requiring filesubmission rather than simple online text.
If someone knows a nifty way to download all OnlineText for a given assignment, that would be great.
I do not find any content-download capability in the gradebook, only the marks themselves -- in any version up through v2.4