What MacBook Air (MBA) is it you have (some specs could be helpful), is it a HDD based model or SSD based and a model year?
If your MBA is a pre June 2013 model, try restart and use Apple Hardware Test (AHT). Instructions here: Using Apple Hardware Test - Apple Support
If your MBA is a post June 2013 model, you will use Apple Diagnostics. Instructions here: Using Apple Diagnostics - Apple Support
If your MBA HDD or SSD has died and is beyond recovery, replace the drive and install from scratch, or restore from Time Machine.
You could try download a fresh copy of OS X and create a USB based installation disc - Instructions here: Create a bootable installer for OS X - Apple Support
You could download the OS X installer (create your bootable installer before running the OS X installer in this next option) and install OS X to a USB drive (8GB or higher needed) creating a bootable USB drive - Instructions here: Starting from an external USB storage device (Intel-based Macs) - Apple Support
Personally, I would create a USB installer first, once done, I would then create a USB bootable drive. (two different USB drives required). I would then restart the MBA with the USB Bootable Drive attached (press and hold down Option key to enter Startup Manager) and select the USB drive you installed OS X onto. The MBA will then start up from your USB drive. Once loaded, see if your original HDD is accessible. If so, it may be recoverable at least to the extent of saving your data from it to a third USB drive. It would also suggest the Boot Sector of the original drive is corrupt since the drive is functioning under a different drive, so a reformat and reinstall may solve the problem.
If not accessible, the drive is likely dead and beyond repair or recovery, in which case a replacement drive would be required. If your MBA is in warranty, or you want to get an out of warranty repair, book a Genius Bar Appointment at your local Apple Store (or Apple Authorised Service Centre). If out of warranty and you're happy repairing your MBA yourself, third party SSD (do check compatibility for your specific model) are available new, or second hand SSD (again check compatibility) from other MBA are being sold on eBay second hand. There are instruction videos online, OWC have several on their YouTube channel.
Hope this helps