My computer is so screwed up; please help!

My computer has gotten really screwed up this past week.

It started with iMovie not responding, prompting me to force it to quit. While the app icon would show that it quit, the window would still be up and the beach ball of death would continued to spin mockingly. So I'd have to restart the computer.

Then it would spontaneously restart when I put it to sleep and have me log in, even though I set it up so that it wouldn't require that under normal circumstances.

Then it would run for an average of five minutes with me doing stuff. Then, I'd do something, and the computer would freeze in place, then restart on its own, prompting me to log in. Once, I even logged in, and in the middle of loading my desktop, it froze, started beeping, and I had to force it to shut down! That happened two times after, the last of which happened in the middle of my newly downloaded Sophos' scan of my memory.

Now, when I start up my computer, I log in, and instead of booting up after I type in my password, the gray apple logo screen pops up and just sits there doing nothing.

With a little luck, my mom's gonna take it to a store tomorrow to be checked, but I want to know if there's any way I can fix it without potentially having to pay a dime, cuz I'm broke. I've tried using Apple's recommendations for starting up safety mode and/or running Apple Diagnostics, but neither of those have worked.

 

That's not gonna work.

First off, I can't back up my data because I can't access it since I can't log in. That being said, I also can't do a fresh install of OS X. If I can't log in, I can't do anything.