I'm at my wits end with this issue… I heard the news of a big Win10 update hitting soon and I wanted to check if I had it already. To my surprise, my system last update was the 6th of October… odd. I'm using Windows 10 Pro x64.
When I try to search for updates, after a long while this is what I get:
Error 0x80080005! I've looked what that means and it's something related to Windows unable to access some files, but I don't know what files. The same happens when using the Windows Store, I can't install anything and I just realized that because I never use the Store.
Next I checked the status of the Windows Update service and it was stopped, when trying to start it from services.msc, I get the error "0x80070002 specified files couldn't be found". 
If I try to run net start wuauserv all I get is that the service couldn't start, System Error, System Error 2 and that the specified files couldn't be found.
I also tried the solution given in this post with no success.
I don't know what else could I try, please help me. 🙁
Best Answer
To give closure to this old issue, what I did is get the latest Windows 10 ISO and install it on top of my current installation, and everything was back to normal.
It's really convenient as it really doesn't override much of my user configurations apart from reinstalling all the default Windows Store apps I manually removed.
I hope this helps somebody!