You can obtain a much larger external drive. For this, it should be an SSD, otherwise speed will be poor. Format it GUID APFS. You can then use SuperDuper or Carbon Copy Cloner (or other similar tools) to make a bootable "clone" of your internal drive on the external drive. Boot from the external drive by holding down the option key when booting to confirm that it starts from the external and everything should look just like it did on the internal drive. All files should be there, except now you will have lots more free space. You can continue to work from the external drive and change System Preferences to make it your regular boot up drive.
At some point, when you have decided to make this change a long term one, you should regularly start backing up your external boot drive that also has all your files. The internal drive can be used for miscellaneous storage after that. It might be useful to keep the internal drive bootable in case you need to return to it for troubleshooting purposes.
If you want to keep all your user files on the internal but boot from the external, that requires more work to configure it that way. The simplest way is to do what I describe above. Making backups regularly of whatever drive your important files are on is essential.
A third approach is to simply get a larger internal drive for your Mac mini, if that is possible.