Macbook Air Hardware Fault
There are only two causes of repeated I/O errors:


A logical failure of the drive due to a corrupted Filesystem 
A physical failure of either the drive or a periphery component (connecting cable). 


S.M.A.R.T. Status is not a reliable indicator for drive failure and is prone to false negatives.  Remember, it’s a firmware function built-in to a drive checking itself. 

Steps:


Boot into recovery mode. Use the repair disk function of Disk Utility to check the boot volume (disk0s2) from your recovery partition. 
Use third party software such as Techtool Pro to scan for bad blocks on your HDD.


Best bet:  

Here is your machine. More than likely, it’s a hardware issue. Specifically, either your drive is failing or the PATA cable is failing. The Apple Hardware Test, even if you were to access, does not test your HDD’s integrity. It would test, however, the input cable to the logic board (the PATA cable) - this is a much cheaper fix.  

Bottom line: You can’t troubleshoot a hardware issue. Backup your data while the drive is still alive. Get a definitive diagnosis from Apple free of charge, and proceed from there.  Once you know exactly what the issue is, decide whether you want to get the HDD/cable replaced by Apple, use a DIY option for a cheaper price, or purchase a new machine entirely. 

BOL.