Do you have access to a second Mac & a USB thumb drive? If you do prep up the thumb drive as a bootable disk.
Then try starting up with it on the Air by holding down the Option key select the thumb drive as your boot disk. If you can get that far first save anything of importance as you may need to wipe the drive. Then next thing here is to free up as much space as you can as I suspect the system can't expand the new files.
The last direction here is to wipe the drive if it still won't fly. Then using your thumb drive set it up as an installer with El Capitan and try again.
OK here goes:
- Using the other Mac open up Disk Utility with the USB drive plugged in you should see it show up. Delete the partition as its likely FAT32 and not workable for a Mac boot disk. Now create a partition you want a GUID & Mac OS Extended (Journaled). I would recommend you rename the drive to something unique.
- Now if you don't already have the OS installer on this Mac download it. Once downloaded make a backup copy and place it in a folder away from the other installer
- Run the OS installer make sure you select the USB thumb drive (16GB or larger). Once it goes thought the process you should be running under the USB drive under the new OS on it. To be sure it works shutdown the system and restart holding the Option key that will get you to the boot manager now select the USB boot drive let it boot up and the last thing here is to copy over the OS installer to it (the backup). Shutdown the system disconnect the USB drive and plug it into your Air.
Boot up your Air now with the USB thumb drive present. Using the Option key like before to select the USB thumb drive. Are you able to get to this point? If you are, delete what you can to make some room & copy off anything important. I would recommend also running a cache & log cleaner like this one: Disk Doctor. After you have cleaned things try running Disk Utility to see if you can fix the internal drives disk & permissions.
Let us know what happens.