Macintosh HD could not be verified completely

Question

Own a Mac air (purchased late 2012) haven't had issues until today when it would not start up.  It chimed, the grey apple appeared. Then nothing. Ran a verify disk and got a message the Hd could not be verified and error: this disk needs to be repaired.  I clicked the Repair Disk which then says disk utility can't repair this disk.
So to complicate this issue a backup has not been done recently and we live out of country and don't have access to an Apple store.  We have an iPad, which I am using now and have an external disk drive but that's all we have to work with. :(
Is there a way to back up the data from a safe mode or whatever Mac calls it?  And could a problem with a file system cause the error? If so, how can that be fixed. Any help is appreciated.
Thanks

Answer 1

So it will boot the Recovery Partition quickly and you show you options for Disk Utility and Reinstall OS X (among others).  That is a good sign.
Try booting in Safe Mode and then running a Time Machine (or other) backup.
(You cannot backup to a Time Capsule while in Safe Boot mode.)
Press the power button and as you hear the bong hold down the Shift key.
If it will boot then run a Time Machine backup.
Time Machine Basics: http://support.apple.com/kb/ht1427
If you could not compete a backup from Safe Mode then skip to the Emergency Data Recovery below before attempting to repair the disk.
Repairing the Disk and Reinstalling OS X.
Boot the Recovery Partition and start Disk Utility.
Click on your system volume (Macintosh HD).
If the S.M.A.R.T status does not say “Verified” then when you are done with these procedures the SSD will need to be replaced.
Repair Disk.
If it did not repair completely the first time, try once more.
Try a Safe Mode boot to backup the data and then a normal boot.
If a normal boot did not work boot the Recovery Partition and reinstall OS X.
Try a Safe Mode boot to backup the data and then a normal boot.
If a normal boot did not work it is time for a clean install.  This will wipe all data from the SSD.  Hopefully by now you have recovered your data.
Unplug any external disks. 
Boot the Recovery Partition. 
Format the system drive using the procedures below, selecting the system disk.
Reinstall OS X.
Boot normally.  If it still won’t boot then there is something still wrong with the SSD or other hardware.
If it boots normally, this time it will ask to reload the data.
Tell it to reload your data from your Time Machine backup or the emergency data recovery disk.
Recover data by booting from an external system disk:
You will need an external drive which you will erase for this procedure. 
(Do not erase your existing backup, even if it is not current.)
Format the external disk as instructed below.
Install OS X onto that external disk, booting the Recovery Partition.
Boot from that external disk (Boot then hold down Option).
Using the Finder drag as many folders as you can to the external disk.
Formatting the external drive:
Boot the recovery partition.
Format (and erase) the destination recovery disk:
          Plug in the drive you are about to erase.
          Select the volume that is NOT your system disk.
          Click on Partition.
          Pull down under Partition Layout to 1 Partition.
          Click on Options…
          Select GUID Partition Table.
          Name: Emergency Backup
          Format: Select Mac OS Extended (journaled)
          Click Apply.
          Click Partition.
Once your system is healthy again be sure to maintain one or two reasonably current backups.
More about backups:
Time Machine Basics: http://support.apple.com/kb/ht1427
Most commonly used backup methods:  https://discussions.apple.com/docs/DOC-3045
Methodology to protect your data.  Backups vs. Archives.  Long-term data protection: https://discussions.apple.com/docs/DOC-6031