How to backup Mac to external hard drive in recovery mode? "No mountable file systems"?
I am assuming you are looking to back up data rather than the entire drive, as your OS image is probably faulty. 

That being the case, you may be able to start up in Target Disk Mode, and then you can browse the files and copy them to another disk as if the HDD in the Mac was just an external drive. There are a couple caveats to this though:


Make sure your Mac is compatible - Given that you are updating to Sierra it is likely that it is.
You will need another Mac to connect the dead one to.
You need a USB-C, Thunderbolt 3, Firewire or Thunderbolt 2 Cable


Connect the computers up, and when starting the broken machine, immediately press and hold 'T'.

All being well it will appear as a disk on the other Mac. You should be able to offload your data from here.

Here is some more info, including compatibility lists: Apple Support