Error turning on PS4: "Cannot access system storage"

Solution 1:

Some PS4s appear to have shipped with a loose or improperly seated hard drive, possibly due to a missing screw. Open the hard drive bay cover and make sure the hard drive is properly seated. If the hard drive screw, marked with the four PlayStation button symbols, is missing, contact Sony for service. (Alternatively, a standard hard drive screw as supplied with most computer cases can be used to secure the hard drive.)

For more information on how to replace the hard drive, see this article.

Side note: The missing screw may be the result of deliberate sabotage at the Foxconn manufacturing plant.

Solution 2:

I was missing my HDD screw. I just installed the HDD and used a standard screw from a old PC to hold it in. After that the system worked fine and updated quickly. No other noticeable problems since. The glossy cover slides off to reveal the HDD. Sony has made the HDD easy to change out and you don't have to void the warranty to check it. Hope this helps