Non-Booting MBA - how to take TM backup?

If the laptop's still alive but just isn't booting all the way successfully then you can follow these instructions to boot the MBA in to "Target Disk Mode", which will let you connect another Mac and get data off the disk. You'll need a Thunderbolt cable.

If there are no signs of life at all then you'll need to unscrew the base, take out the SSD and use one of these to connect it to another Mac like any other USB drive.

Alternatively, you could take it to an Apple Store (or authorised reseller) and ask nicely if they'll help you get the data off before they try to fix it. Take a big enough external hard drive with you if you try that.

 

The other option is to find a working copy of OS X and use something like SuperDuper! to copy it to an external drive.

You can then boot from the external drive and remove some files (This is assuming your hypothesis is correct, which I am not sure it is.)

To boot from an external drive hold the option key right after the chime and then arrow over to the other volume.

 

Or, if it can boot to Recovery Mode, you can use Disk Utility from there to clone the internal drive to an external.

 

Get an USB installer disk and select that disk (hold option) while booting.
Then sort it out from the utilities.