The interpretation of Quora’s T&C I’ve seen is that you can backup/archive your own content (since you retain full rights to it), but nothing by anyone else.
Quora does not provide this functionality. Of course, Quora does not provide a whole lot of functionality.
There are several third party solutions that have come and gone. The most recent I’ve seen is:
Stefan Delic’s answer to Does Quora have any plans for a feature to allow listing and downloading of a persons own answers in bulk? (as plain text)
The one I’ve used quite happily is: