you have 2 options
continue to ignore the emails as it looks like you have done so far, this is only asking for troubles
Or
try checking log directories and clearing out old rotated logs
check your backup isnt also being stored on your main harddrive and either copy them to another drive or ideally off the server then delete from local disk
how big is your drive and how much are your sites using of it? it may just be a plain and simple, your website files have filled and you have truly outgrown your disk (so you need a bigger one or a new server)
one other thing to check if you have stuff like cron jobs running etc and if you dont use the server for email is to check and delete mail stored on the server, for example cronjobs will send mail to the account that they run as so checking webmail in cpanel will show this and hopefully you can delete them
to check log file size
du -h --max-depth=1 /var/log
to delete files
rm -f /var/log/logfile.log
anything that has extension .gz or .(number) are more than likely safe to delete, however you may want to keep the most recent ones just in case you need to check your logs, perhaps download them from your server to your machine before you delete