Could not communicate with Ghost or You don't have enough free space in /var/cache/apt/archives/

Ok, so the issue happened because there was a massive 5G log file, which blocked all the space.
After deleting log file the issue was solved.
Here is how to do it - use main user, not ghost-mgr.

cd /var/www/ghost/content/logs
rm https___YOURSITE_production.log

I also looked in this thread to fix logrogate configuration for self hosted Ghost installs.