El Capitan Install Failed - Now getting kernel panic on login

I have been running the most recent version of Yosemite on a 13inch 2014 Retina MacBook Pro. When I installed El Capitan, halfway through it told me the install could not complete due an error extracting the Essentials.pkg file. It advised me to restart and I did, but now when I try to log into my account, white text highlighted on black appears at the top, saying an operating system is not yet set. The computer then crashes.

I started in recovery mode, but the option to reinstall El Capitan tells me I have to restore. This is a problem, as I don't have a backup available of my macbook's SSD. Although I can get back most of the things on there, some things would be permanently lost, so I am reluctant to follow this. I am also presented with disk utility, but I am aware the option to repa disks is no longer used in El Capitan and I am worried that it will worsen things. My Macintosh HD drive is also fully blue, which I'm worried is showing I have lost my data.

I did several diagnostics tests, which found nothing, and also made it to the guest safari page, which is using San Francisco, suggesting some of my installation is complete. However I found no solutions on the apple site.

I've read about a tool called target disk mode, which I'm thinking I could use to transfer my macbook's files to my father's Mac. However, I don't have a thunderbolt to do this (if I went to a Genius Bar, do you think they could do this instore?)

If anyone could help me on how to otherwise retrieve any personal files left on my hard drive and get my MacBook running again, it'd be greatly appreciated. While my music library is replaceable, I had to keep files for my music production software Caustic on the SSD, and I don't want to lose them.