Set a shortcut to toggle bluetooth on or off

Is there a way to do this without installing extra packages?

You can control whether your bluetooth signal is enabled with rfkill. Wrapping this in a little Bash conditional allows you to toggle the state easily:

#!/bin/bash
if rfkill list bluetooth | grep -q 'yes$' ; then 
    rfkill unblock bluetooth
else
    rfkill block bluetooth
fi

You could save the above in a script file anywhere (e.g. ~/bin/toggle-bluetooth) and make it executable (chmod +x FILENAME) to be able to bind that command to a keyboard shortcut in the system settings.

Alternatively, you can put it in a single line bash command and directly paste that into the shortcut:

bash -c "if rfkill list bluetooth|grep -q 'yes$';then rfkill unblock bluetooth;else rfkill block bluetooth;fi"