The root filesystem is full. Follow my advices:
https://trisquel.info/forum/problems-synaptic-and-apt-get-update#comment-64463
That said, given the size of that partition (5.7G), you will face this
problem over and over again. The default install chooses 14G for that
partition. I do not get how your first disk is less than 20G large. Is that
correct? I see that you have a large external disk (500G). I assume you keep
most of you data on it, don't you?
If so, you should enlarge /dev/sda5, taking space on /dev/sda7. In fact, you
could even get rid of /dev/sda7 (user files would be on the same partition as
system files): ether you reinstall Trisquel (choosing a custom partitioning
schema when asked) or you edit the partitions from a live system (using
GParted for instance) and adapt /etc/fstab accordingly (see
https://trisquel.info/forum/cant-use-gparted-either-livecd-or-installed-trisquel).
In both cases, you first want to backup the 978M of data in /home (to your
external drive for instance).
