Can't restore from backup or reinstall OS X

So last week my screen was frozen on the screen saver. I turned it off and back on. It took forever to start up and brought me to a OS X Utilities screen.

First, I did disk utility. There was an error at first but running first aid fixed it. Rebooted and still got utilities screen.

Then I tried to restore from a time machine backup. It allows me to select a back up but after I click continue it brings me to the Select a Destination screen and it says "no time machine backups were found".

I would prefer to restore from a backup but I'm having problems even completely just reinstalling OS X, when I try to choose recovery disk it tells me that is it locked. I have tried quite a few things to unlock with no luck.

Any idea what is wrong or what I can do? Any help is greatly appreciated. I have been trying for about a week to figure this out. And using a tablet or my husband's computer with windows in the meantime and it sucks.

Thanks!

 

Ok, that's what I figured it was but wasn't sure since it was showing up as having no problems on first aid. Damn, well since it's about 10 years old now looks like it's time to retire it finally. Thanks for your help.

 

Quick question for you. I have back ups with time machine and also an external hard drive that I had to format for my mac. Is anything retrieveable without a mac? I can't afford a new one right now and don't want to keep putting money into this one as it's 10 years old now. The only other computer we have in the house right now is my husband's laptop which runs Windows. Thanks!