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.