Failed to add /run/systemd/ask-password to directory watch: No space left on device?

As discussed in a Red Hat bug report, it turns out that the Crashplan back-up service is the most likely culprit. It uses many inotify watches and, eventually, eats them all up.

The immediate fix is to run:

sudo -i
echo 1048576 > /proc/sys/fs/inotify/max_user_watches
exit

to make more watches available.

The long-term fix is to edit the file /etc/sysctl.conf to include the line:

fs.inotify.max_user_watches=1048576

I'm not reputable enough to comment on the accepted answer, but I wanted to say that it's by no means limited to CrashPlan. Dropbox and other file sharing platforms use inotify watches per inode to detect when an upstream sync needs to occur. Malware detectors may have watches on directories. Other backup tools besides CrashPlan might, as well.

To see what's consuming inotify watches, use lsof:

sudo lsof -K | grep inotify | (less||more||pg)