I am supposing this will not work - simply using dd to copy the booted disk /dev/sda for example to a file on an external hard drive. This would not be a good back as too many system file would be missed. right?
Right. You can’t get a correct image of rw mounted filesystem.
Is there an easy way to create a very minimal linux that runs out of memory or that would live in a very small partition, that would be capable mainly of dd ing the working partition to a file on a USB?
You can boot off usb stick. Or grab a copy of clonezilla and boot off that memory stick with all the tools you’d want for drive or partition copy/restore
I think it’s good enough. Shut down as many services as possible if you want to be safer. In the end it would just be like booting after a power failure.
it would just be like booting after a power failure.
That level of confidence is EXACTLY what I want when I’m restoring a backup.