2010 MacBook Pro stuck in update loop

Hi all. I have been trying to update from Yosemite to High Sierra and have got stuck in a loop. After doing some research I was able to restart into disk utility and I tried first aid but it says that disk utility can’t repair the disk and it needs to be reformatted. I have some files I really want to recover first. I bought an external drive to back up to but it wasn’t compatible with Yosemite 🤦‍♀️ I’d only upgraded from Mountain Lion to Yosemite earlier this year so thought I’d be safe.

Ive tried restarting from Disk Utility but it asks me what start up disk and that fails (sorry it was a few days ago now and I can’t remember exactly what it said). I tried the different different key plus R combos for restart and then reinstalling the OS but all have failed. When it tries to do the oldest version (Mountain Lion) it says it can’t because a more recent version has been installed. When I tried to go back to Yosemite it says the file is missing and when I try to go to high Sierra it fails too.

I’ve attached a picture of the disk utility error I was getting.

Is there any way for me to get in a get this files? It’s mainly photos that I’d like to save.