Not seeing hard disk in Disk Utility in Internet Recovery Mode System specs: 2010 MacBook Pro Mountain Lion. 

It all began with the grey screen and spinning wheel..... there were no prior symptoms indicating problems, plenty of available memory... etc.

I have fsck'd in more ways abd more times than you might be able to imagine
I have used Disk Utility from the Recovery HD in the start up manager and it reported permissions problems.... but also said the hard disk was okay.

Now there is no option to boot from any volume in the start up manager. Only Internet Recovery Mode works, which is how I am writing this.

I clicked on reinstall Mountain Lion, but it reports that it can not and that I must contact Apple Care.
There is no Apple Care where I am located, and I do not have a phone that functions here either.

I looked in Disk Utility and all I see is the greyed out superdrive and then Disk0 base system OS X.
I do not have an external hd or another Mac, and I do not have backups of my files and I would prefer to keep them.

Please let me know why OS X Mountain Lion will not install.
Based on the evidence I believe your hard drive has failed, and needs to be replaced.
You don't have to take it to Apple, there may be authorized repair centres near you.