How much free storage do you have at all? It is not not save to operate a mac, it the free storage is critically low.
Quit all applications, empty the system Trash, then delete files, that you can sacrifice because you have backup copies.
Restart the Mac to free memory, then try again to ope files to an external drive.
Do you have access to a second computer, perhaps you can can borrow a mac from a family member? Then you can try to boot your mac in target disk mode; this way another mac can see it as an external disk and copy data from your mac to a backup disk, see this document: https://support.apple.com/kb/PH3838?viewlocale=en_US&locale=en_US
Once you get your Mac back to normal, avoid in future to let the free storage drop below 20GB or so. And make regular backups, for example with Time Machine, so you can recover from similar situations more easily.