Sudden proliferation of 'Too many open files in system' errors. Can't do backup
I have several suggestions that I'll put into two different posts.

Here's the  first one:

This seems to help with practically everything, so why not try it.

Open your Disk Utility app, and after selecting your main hard drive (or partition), follow the steps in the picture below. This operation will attempt to find and repair any disk and disk-permission errors.

Please note that step 1(b) probably will not work unless you are booted into the Network Recovery Mode, or running off a different hard drive (not the one you are trying to fix). If running step 1 reveals a lot of errors then doing 1(b) would definitely be recommended.

The easiest way to do this is to borrow someone else's Mac and connect your computer to it in Target Disk Mode. This will allow you to run Disk Utility on their computer thus letting you perform step 1(b) to your hard drive.

If you don't have another computer to borrow, your will need to put you computer into Network Recovery Mode (aka. OS X Internet Recovery). Please see the OS X Internet Recovery section in this hyperlink before proceeding with this fix. Also note that Network Recovery Mode is different from Recovery Mode; the regular recovery mode will not help in this case. To start your Mac in the Network Recovery Mode, please see this link.

If none of the above are available on your computer, or Network Recovery simply takes too long on your internet connection, there is a third option. You can make a Bootable OSX Recovery Drive from a USB stick as described at the bottom of this page from Apple.

Hope this helps! :)