Need help backing up Macintosh HD to a WD Passport for Mac

Do you happen to have another Mac handy or can borrow one? If so, you could try Target Disk Mode so that you can access your Mac Pro's drive from the other Mac. From there you can make a clone copy or just copy any files from it.

Ref: Share files between two computers with target disk mode - Apple Support

If you don't, then you can try starting up in Safe Mode to see if you can, at least, get to the Desktop.

Ref: Try safe mode if your Mac doesn't finish starting up - Apple Support

Hmm, that doesn't quite make sense. A normal startup would go something like the following:

1. A short time after power is applied you would hear the startup chime.
2. This would be followed by a black screen to be replaced with a screen with the Apple logo.
3. Immediately after this, you should see the progress bar.
4. If all goes well, you would then see the login screen or the Desktop if you elected to not use the login option.

You posted that you are able to enter a password. Are you referring to the one you would enter at the login screen or one that you would get before seeing the Apple logo?

If you have driven the free space down to zero, the Mac can't start. You are going to have to another Mac and Targeted Disk Mode to recover your data now. Either that or remove the disk drive and mount it in an external enclosure to recover the data that way.

Try doing a safe boot. It might free up enough space to get your Mac working again.

Sorry. I was hopeful that the easy solution would work for you. Now it appears you are left with the suggestions Tesserax has given. Hopefully, one of them works for you.

Ok, thanks. That makes sense for me now. Again, it looks like you will have to try the Target Disk Mode option to resolve this.