Totally agree with altrenda, boot to target disk mode (hold "T" on startup) and connect via firewire to another computer.
There is no firewire port on a Macbook Air. TDM isn't an option.
The machine should have a recovery partition if it is running any OS above 10.6.8. Connect an external HD via usb. Turn on the computer holding the option key. There should be 2 volumes listed if the machine is running an OS newer then 10.6.x. One will be listed as "Macintosh HD" or similar. Other partition will be listed as "10.x recovery". Choose the Recovery partition.
When booted to the recovery partition, you will be presented with some selections. Choose Disk Utility (should be one of the selections presented). Once in Disk Utility select the volume "Macintosh HD" or similar. Once that volume is highlighted select either the "restore" tab or select "new image" along the top right row of buttons. Your selection will depend on if you just want to make an image file or if you essentially want to clone the volume. Choose "restore" if there isn't anything that is wanted on external HD.
All of this is assuming the external HD is partitioned as GUID and the partition map is Mac OS extended (Journaled) (*Can't use an NTFS volume and FAT is only good for file sizes less then 4GB). If the external is not set up correctly then do a search for partitioning and formatted a windows drive for Mac or something similar. I really don't want to get sidetracked with explaining that process as well. But you are in Disk Utility already you'll want to stay where you are to also prepare the external HD. Also, make sure any data already on the external drive is backed up or not needed.
Using the restore method (to clone) drag the volume "Macintosh HD" or similar to the "source" box. Drag the external HD to the "destination" box. Once that is done, click the restore button in the lower right. It will prompt you about data loss and such, but as long as you have a good understanding of the source and destination just click okay. You'll see a progress bar until the process is finished. Once it is complete, there will be no indication except the progress bar will be gone and the external HD will more than likely have taken on the name of the source volume.
If creating a disk image. Click "new image". A finder window will open asking you where you want to save the image. choose the external HD. No need to change the name of the image file as the default name should be the "volume name.dmg". Click save and the process begins. Watching the progress will be nearly identical to the restore procedure listed above.
When either method is completed. Just connect the external to another booting Mac. If the restore procedure was used, the volume will just appear mounted on the desktop. You should be able to freely browse the directory to verify everything is there. If you used the new image procedure, you'll have to go into the external HD and double click on the "Macintosh.dmg" file. It may go through a lengthy verification process before mounting. This of course can be skipped by clicking skip on the verification window. Once mounted the volume will appear on the desktop as "Macintosh HD" or similar.
