Keep getting Blue Screen of Death, even after formatting hard drive

A while ago I was dusting out the computer case. Around this time trouble started. The computer failed to start and would often come up with the message

A required device isn't connected or cannot be accessed Error code: 0xc000000e

I was able to boot from a live USB and restore from a System Recovery Point. However the problem kept coming back. It would be good for about a week then startup would fail. I decided to backup everything and reinstall the operating system. Windows seemed to install ok but after the first restart it went straight back to the BSoD.

What does this mean? Does this prove it's a hardware problem? When things are half broken like this and sometimes work, what is a good way to diagnose the problem?

The computer has two internal storage units: a SATA II hard drive and a SATA III SSD.