Computer starts up to gray screen, in recoevery mode getting "Open Error 5: Input/Output error on System/CoreServcices/ManagedClient.app/*
You need to specify what year/model your Mac is, and what operating system came with the system.

Let's assume that "Lion" came pre-installed and you have the option of a "Recovery Partition". You need to first format your hard-drive, removing everything (making sure that you have a back-up!!) and then:

1) Switch off your mac

2) Book your Mac and press and hold the ⌘ (Command) key and R key immediately; after the first chime sounds. You must keep holding down both the Command and R keys until you see a window with “OS X Utilities” across the top in large text. This should enable you to do a fresh re-install. 

The second way to access the recovery partition is to re-boot your mac and hold onto the (options) key. 

If you do not have a recovery partition on your hard-drive you need to seek someone who has a Mac with lion+ on and download the OS X Recovery Disk Assistant v1.0 and placing a new image onto a memory stick, and following the steps above with the memory stick. 

If your Disk cannot be repaired, then, there maybe an issue with the disk itself. Or, you could try and "Erase" the disk and doing a fresh re-install. Again, you need to TRY the best you can to ensure you back up all the data before trying any of these steps. But, it seems to me (personally) that this may not be an option for you. 

I hope this helps :)