MacBook Air Hard drive crashed (2013 Model) Is it possible to replace the hard drive for a 2013 MacBook Air? Where can I purchase a new hard drive?

If not, is it possible to run it with a portable hard drive?

I purchased it in 2014 (refurbished). So, it's probably 2013 model and not under Apple Care. Apple customer service told me that my hard drive has crashed.

Does Apple sells hard drives?
You can replace the SSD (Solid State Drive, it is not a hard drive with spinning disk)!

.... with a model from OWC yourself if you know your way around embedded hardware. 

Apple does not sell parts to individuals, but an AASP or APSP (Apple Service Providers) can replace it for you with an Apple part. 

The SSDs are not compatible with m.2 or mSATA, you have to use the specific MacBook Air SSD for your model. Major MacBook Air releases also have different SSDs due to incompatible upgrades (protocol upgrades and electrical upgrades).

Sites like iFixit have guides that can assist you with this.

See some instructions here https://everymac.com/systems/apple/macbook-air/macbook-air-faq/macbook-air-mid-2013-how-to-replace-upgrade-ssd-storage.html