Is my iMac Hard Drive Toast?

Hi all! So about a year ago my backup iMac (2010) starting shutting off on boot up. The bar would start progressing on the boot up screen but the machine would shut off before it would complete boot up. We were moving at the time and didn't have time to figure it out. A year later, Im wiping the dust off this old machine and trying to figure out what we need to do to get it running again.

Last night we went through a myriad of suggestions from the internet on bootups and by the end of the night ended up with a blinking question mark file folder on startup.

I should have taken better notes of our steps but I will try to remember and describe below. P.S. We are not super tech savy - just a lot of googling and trying things!

1. Booted up in recovery mode to get to the disk utility where we tried running First Aid. This resulted in an incomplete first aid run with a S.M.A.R.T error which we read could mean the OS is malfunctioning or the hard drive has failed.

2. My husband then did a reformat on the machine. He says he did a partial reformat which didn't change anything, then came back and did a full reformat.

3. After the reformat it took us to a rotating globe and a progress bar for Internet Recovery. Internet Recovery progressed the whole way through (about 30 minutes) and then went to a blank white screen with gray apple logo. And it just sat there. We tried this whole Internet Recovery process again with the same result. We left the blank apple screen up when we went to bed thinking it just needed some time to do something. It was still there this morning.

4. Now when we try booting up we get the blinking question mark file folder. I know this is because wiping the system also took off the OS and the computer has nothing to point to when starting up. I have the OS installation disc and have tried starting up with it inserted to see if it will do something.. nothing.. It sounds like it is reading the disc a few times then spits it out. I read to hold down the C button on start up with the disc inserted which doesn't do anything differently.

5. I also read to try resetting the PRAM (CMD Option P R on start up) but this does nothing different either (tried this before and after the reformat)

So, does anyone have any other troubleshooting to send my way? Or does this sound like the hard drive is toast, as I have been thinking?

If toast, what do I need to buy to fix it?? We have never changed a hard drive in an imac but we are real good at following directions (thanks youtube :) So we'd like to give it a try.

Thanks in advance for your advice!