This got me worried, but checking drive space, I found that all the drives had plenty of space, and there were no shortage of inodes (a common reason for the message). I rebooted (the shutdown command also resulted in a "Not enough space on device" message, but it did restart), but this did not solve the problem. I then decided to watch the log (tail -f /var/log/messages) and spotted the following message:
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tail: inotify resources exhausted tail: inotify cannot be used, reverting to polling
There are some instructions for increasing the number of inotify watches on your system on the Crashplan website
Update: I spoke too soon - the ntpd service is still continually restarting (once every minute or two). No more "Not enough space messages", though!