I'm really hoping this doesn't get downvoted as while this issue is severe...hope is not lost. So please read through it carefully.

What appears to be occurring, and I hope those with more IT experience than I do can add further to this, is Anthem is corrupting the PS4's file management system. THIS DOES NOT AND SHOULD NOT MEAN YOUR CONSOLE IS BRICKED

All of your data is still on the HDD or SSD currently in your consoles. Just temporarily inaccessible. HOWEVER, this issue is easily rectified.

You must boot your console into Safe Mode by holding the power button on the console until you hear two beeps. It must be the power button on the console itself, you cannot boot the console into safe mode via the controller. Once in Safe Mode you should be able to select an option labeled "Rebuild Database". This will fix the issue.

I had a problem exactly like the one going on now a couple years back with the Kingdom Hearts 1.5 2.5HD Remix. It might be a different game but the issue was exactly the same as was the way of remedying it. I really hope this helps those who still wish to give Anthem a chance despite the technical disaster that it is on all systems. I'm hoping I'm right about this. I'd rather not hear of people throwing money at a new console despite the "bricked" one being completely recoverable.
 

EDIT - DO NOT SELECT INITIALISE! This option will completely wipe your HDD/SSD. The option you want, as I have stated, is "Rebuild Database". I'm only mentioning this as an advised precaution.
 

EDIT 2 - It appears some people are misunderstanding the intention of this post. The goal is to provide the users who may have seemingly bricked their systems of a solution as well as informing those still playing Anthem on the PS4 to a solution should it ever happen to them. This is in no means encouraging people to suck it up and continue playing on the PS4. My honest advice to this whole situation would be to stop in the off chance something more severe rears its ugly head. But if you're intent on trooping on forward then I'm not here to scream at you or judge you. It's your call. Just be aware of the risks that lie ahead. I'm just glad this post got as much attention as it has done.