My mac book air doesn´t start, only the apple logo and the beachball spinning after restart

Try running Apple Hardware Test on the machine (it's one of the discs that came with the machine.) Also, if it's freezing by showing the spinning rainbow cursor (beachballing) you may have a failing hard drive. That wouldn't be uncommon for how old the machine is.

1. Restart.
2. Press F2 key couple of times
3. Reset PRAM. http://support.apple.com/kb/PH4405
4. Reset SMC. http://support.apple.com/kb/HT3964
Choose the method for:
"Resetting SMC on portables with a battery you should not remove on your own".
If this does not help, contact Apple.
Cheers

Recovery System

OS X Recovery includes a built in set of utilities as part of the Recovery System. To start your computer from Recovery, restart your Mac and hold down the Command key and the R key (Command-R). Press and hold these keys until the Apple logo appears. This indicates that your Mac is starting up. After the Recovery System finishes starting up, you should see a desktop with an OS X menu bar and a "Mac OS X Utilities" window. Note: If you see a login window or your own desktop and icons, it's possible that you didn't hold Command-R early enough. Restart and try again.

The Famous SPINNING BEACHBALL. It should stop after a few seconds, if not BackUp and then Reformat your HD and start over.

The time bar may be telling you that there's damage to the hard drive, or perhaps just your hard drive's main partition.
If your Mac is running OS X Lion or Mountain Lion, you should have a recovery partition with Disk Utility on it. Restart your Mac while holding down Command-R to boot into this instead of your regular partition. Follow Apple's instructions on how to use Disk Utility to see whether the hard drive is damaged or just corrupted (and potentially corrupted).

The "beachball" and sounds do typically mean that your drive is failing. Your plan to back up is the right choice.

You should be able to replace the hard drive yourself. Opening the case and replacing the drive is exceedingly simple on the mac pro.

Have you tried booting from an install disc? Or in safe mode?

When troubleshooting a Macbook Air, there is a checklist from Apple Support you need to try.
You are welcome to do so after first trying these steps first.

• Check the Magsafe power adapter, to make sure it is free of debris and is seated properly.

• If with decent charge, try to start the Macbook Air, with only the battery.

If you get power from either one of these issue resolved- otherwise follow the steps provide more in detail from Apple listed in the link below.
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Apple Support Link:
Troubleshooting Macbook Air (additional steps)
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Such power issues with Macbook Airs as well as Macbook 13" and Macbook Pro 13" are about the same and there are no workarounds, so make sure to follow those procedures completely for a full assessment of your Macbook Air's functionality.

Hope this helps!

Regards,

-Mactechtrainer

Restart your mac:
Safe Boot: Hold down the shift key after the chime until the Apple logo appears and the gear is moving.
It will take awhile but you will be asked to login.
Go to your Applications folder> Utilities and launch Disk Utility.
In the left side column, highlight your drive and repair permissions.( button down below)
Empty the trash.
Restart your mac

To reboot a Mac laptop, go to the Apple logo in the top left hand corner, and click "Restart". If your computer is unresponsive, press and hold the power button for about ten seconds.