Drive died?

Last night my PC just completely froze on me, so I had to do a hard shutdown by hitting the power button. When I tried to reboot, it gave me those recovery options—Safe Mode, Last Known Good Configuration, or Start Normally. I tried a normal boot, but the Windows logo just spins for a bit before it kicks me out. Then the computer restarts itself, hits a black screen, and tells me "failed to boot, insert disk."

Every time I try any of those other recovery options, I get the exact same result.

I even tried booting from a Windows installation disk, but when I get to the part where it asks me to select a drive for the installation, it can't find any disks at all.

Maybe try checking your BIOS first to see if the hard drive is even being detected? If it’s not showing up, I’d suggest swapping out the SATA cables—just to be 110% sure you're actually dealing with a dead drive and not just a faulty cable causing all the headaches.

It isn't dead, because if it were, you wouldn't even see the operating system logo spinning. However, it is entirely possible that the drive itself has sustained some physical damage.
The most prudent course of action would be to install a secondary drive, set up a fresh OS on it, and use that to attempt a repair of the existing installation still sitting in your machine... alternatively, you could take the drive to a friend's place and let them run some diagnostic tools to "comb through" the file system.
In any event, start by checking the BIOS and following all the advice provided by Larry Ortiz5. It is also quite plausible that your CMOS battery has simply failed.