Mac won't boot. Infinite loading screen. OS X reinstall not working.

13" MacBook Pro, Early 2011, 2.3 GHz i3, 8GB RAM, 320GB HDD.

About two days ago, I left my computer paused on Stardew Valley for around 6 hours (fans were at full blast) and when I came back to it, it was acting very weird. Corners weren't working, gestures registered a minute later if at all and my usb mouse wasn't working. After a couple minutes trying to fix it, I ended up just holding down on the power button. Since then, I haven't been able to turn it on.

When I tried turning it on, the progress bar loaded very slowly and when it finally finished after about 5 minutes, it just stayed like that. Since then, I've tried everything I know but still I get the same thing - 5 minutes for the progress bar to load and stay like that.

What I've tried:

• Waiting for hours after progress bar has loaded.
• Resetting PRAM
• Resetting SMC
• Trying to boot into safe mode
• Repairing "Macintosh HD"
• Clearing up space on "Macintosh HD"
• Reinstalling OS X

I retried every single one of the steps above multiple times. When I tried reinstalling OS X, I ran into a space problem so I cleared up some space and I tried to boot again to make sure the lack of space wasn't the problem. It didn't work so I started reinstalling OS X. After downloading, the Mac restarted and started the install, but the progress bar never got past about a tenth of the way through before coming up with an error: "The OS X installation couldn't be completed". I managed to save the log but it was too big to fit on pastebin so I had to use Dropbox - Installer Log 4-Nov-2017.txt

I can't backup my drive right now so I can't attempt a clean install. Any chance someone could look through the log to find out why the OS X install isn't working? Sorry I won't be replying for a while, it's 11:30 in the morning and I've been up for about 24 hours now - I need to go to sleep. Also, sorry if there's anything that doesn't make sense in this post, I've fallen asleep more than ten times while typing this out.

Things that might be important:

• I replaced my battery about a year ago
• I upgraded my RAM about two months ago
• I had 6GB of space left over on my machine when this happened