I was wondering if it was possible and or if someone had attempted it to stuff a PC inside of a PS4 shell?  I'm sure one of those micro PC motherboards would fit but I have no idea about the rest of the components.  I'd be interested in buying a non-working PS4 unit and putting a PC inside of it.  

One word. Heat.

I cannot see running any decent PC parts in a tiny PS4 case. Even if everything fit, which odds are it is gonna be a tight squeeze, even with low profile parts, the heat is going to be a major concern. Consoles are specifically designed for months/years by a professional design team with certain parts to make sure they run efficiently, and even then sometimes they still have issues and overheat. 

This is a very good point.  I'd likely use the on-board video so that saves me some heat from a separate video card.  I am not opposed to installing some low profile / smaller fans on the case to help with the cooling.  

I don't own a PS4 at the moment so I'm not able to get any measurements to know what fits and what doesn't.  My biggest concern is likely the power supply...  

*edit*
Console size...
2.09 inches (h)
10.83 inches (w)
12.01 inches (d)

Yeah heat would definitely be the biggest issue. Even with the designers working for months/years planning out the perfect (often custom) parts to make it all work and fit in a low profile casing, they STILL sound like you're running a hairdryer sometimes, and get so hot that some games will blue screen a well used console.