it should be possible to install some flavour of Linux on your old mac mini and use it as a NAS with external drives. Looks like its possible to use the mac HFS+ file system with Linux but journaling needs to be switched off first.
I am no mac guy and I like to tinker with stuff and avoid e-waste, so this would be the thing I would try. This level of low-level tinkering with software and possibly spending a few days getting it working is not for everyone, obviously.
NAS... Synology or QNAP maybe? After all, a NAS is just a normal computer in a special form factor with special software to make the config easy. Any old computer with a specialized software (e.g I run openmediavault) should do.