The OP asked:
"Can I run a mac entirely off an external hard drive?"
Yes.
Of course you can.
This is child's play on a Mac.
I've been booting and running my late 2012 Mac Mini from an SSD (in a USB3/SATA dock) since the first day I took it out of the box in January 2013. Still runs great.
Get either a "ready-to-go" USB3 SSD such as this:
http://www.amazon.com/exec/obidos/ASIN/B00ZTRY532?tag=delt-20
or...
Get a "bare" SATA SSD and either a USB3 enclosure (make sure it supports UASP), or a USB/SATA dock, or even a USB3/SATA adapter dongle like this:
https://www.amazon.com/Sabrent-2-5-...478&sr=1-2-spell&keywords=sabremt+usb3+to+ssd
Then, set up the SSD with the OS, applications, and your "basic" accounts.
By "basic", I mean leave large libraries of movies, music and pictures on another drive (they don't need speed).
The Mini will run much better.
If the original drive inside was a platter-based HDD, you won't believe the increase in performance until you try it.
It will run far, FAR BETTER THAN NEW.
Hmmm... as others mentioned above...
What makes you think the drive has failed?
What happens (exactly) after you press the power on button?