How to backup Mac to external hard drive in recovery mode? "No mountable file systems"? I upgraded my mac to High Sierra without backing up my data. Then my Mac wouldn’t login, it gives me a black screen then it restarts. 

So I decided to reinstall the software, but first I need to back up my data. I’ve been trying since last night. I put Mac in recovery mode, then I created a “compressed” disc image to put it on an external hard drive. I wanted to make sure I can open it before I reinstall and in the process lose all the data. I put it on a friend's Mac, and it keeps saying “no mountable file systems” and the dmg won’t open. The dmg file has a size of 500 GB. 

Is there any other way to backup my data, given that I can't login into my Mac after the High Sierra install?
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