Exim mail queue can get out of control. If your var is on a separate smaller sized partition it can quickly run out of space.
We can see what's using up all that space via:
du -sh * | sort -n
and as with Exim, our culprit location should be:
So to quickly clear out the exim mail queue we do the following:
cd /var/spool/exim/;rm -rf input msglog;/scripts/eximup --force
And that should do it.