what is the meaning of the "root file system full: empty trash" message?

there should be plenty of space:

nicholas@mordor:~$ 
nicholas@mordor:~$ sudo df -h
Filesystem                 Size  Used Avail Use% Mounted on
tmpfs                      1.6G  2.6M  1.6G   1% /run
/dev/mapper/vgubuntu-root   72G   68G     0 100% /
tmpfs                      7.8G   16K  7.8G   1% /dev/shm
tmpfs                      5.0M  4.0K  5.0M   1% /run/lock
tmpfs                      4.0M     0  4.0M   0% /sys/fs/cgroup
/dev/sda2                  512M  7.9M  505M   2% /boot/efi
tmpfs                      1.6G  1.7M  1.6G   1% /run/user/1000
tmpfs                      1.6G  892K  1.6G   1% /run/user/1001
tmpfs                      1.6G  940K  1.6G   1% /run/user/1004
nicholas@mordor:~$ 

I've only ever looked at /dev/sda2 and not the "root" above with 100% use.

Having just launched the alpine e-mail client I went to save (meaning download) five e-mails in my inbox, I got:

                                       [Error writing scratch file: No space left on device]
? Help                < FldrList            P PrevMsg               - PrevPage          D Delete              R Reply               
O OTHER CMDS          > [ViewMsg]           N NextMsg             Spc NextPage          U Undelete            F Forward            

which is very odd. Running du -sh is taking quite some time, here are the results so far:

root@mordor:/# 
root@mordor:/# pwd
/
root@mordor:/# 
root@mordor:/# du -sh *
0   bin
169M    boot
4.0K    cdrom
16K dev
17M etc
54G home
0   lib
0   lib32
0   lib64
0   libx32
16K lost+found
8.0K    media
4.0K    mnt
4.0K    opt
du: cannot access 'proc/114024/task/114024/fd/4': No such file or directory
du: cannot access 'proc/114024/task/114024/fdinfo/4': No such file or directory
du: cannot access 'proc/114024/fd/3': No such file or directory
du: cannot access 'proc/114024/fdinfo/3': No such file or directory
0   proc
34M root
du: cannot access 'run/user/1004/doc': Permission denied
du: cannot access 'run/user/1004/gvfs': Permission denied
du: cannot access 'run/user/1001/gvfs': Permission denied
du: cannot access 'run/user/1000/doc': Permission denied
du: cannot access 'run/user/1000/gvfs': Permission denied
6.0M    run
0   sbin
6.3G    snap
4.0K    srv
0   sys
13M tmp
7.0G    usr
6.9G    var
You have new mail in /var/mail/root
root@mordor:/# 
root@mordor:/# whoami
root
root@mordor:/# 

which I ran as root as I was in the / dir.

I'm thinking that the boot partition is incorrectly configured and mixed up with the root directory somehow(?).

Solution 1:

5% (by default) of the filesystem is reserved for cases where the filesystem fills up to prevent serious problems. Your filesystem is full. Nothing catastrophic is happening because of the 5% buffer. root is permitted to use that safety buffer and, in your setup, non-root users have no reason to write into that filesystem.

If you have daemons that run as a non-root user but that need to manage files in that filesystem, things will break. Time to find more space.