I have resolved this problem, but I wanted to post this up for anyone having the same issues. I'm not saying this will work for you but it could help. There are a lot of unanswered problems and bug reports in these forums right now, it's up to the player to share their resolutions until we can get a patch to fix some of these bugs.

I purchased the digital version from the Playstation Store after I gave up on my physical copy ever working. After getting it downloaded and starting a new character I ran into an issue right from the start. The game thought there was not enough space on my hard drive to accommodate save files. As a consequence, I could not save the game at all. Every time the game tried to autosave or if I attempted to manually save, I got the same prompt telling me that there is not enough space and I needed to make some. I had over 250gb of free space on my PS4's hard drive after downloading Fallout 4, there is no reason to expect that this game requires more than that to create save files.

I tried making more space by deleting some of my applications first, that did not help. I tried to play on a different profile, still got the same issue. Rebuilding the database also did not help. For some reason the game did not believe it had enough space.

I ended up re-initializing my PS4 to wipe the hard drive and then reinstalling Fallout 4. This fixed the issue. Just deleting and reinstalling Fallout might have solved it as well, but I wanted to make sure I had a clean drive before I spent the time to download the game again. I've been able to play and save the game without issue. If you decide to re-initialize your PS4, be mindful that it will wipe everything of your PS4 including games and apps, saves, and your profiles. Only the current firmware version, which is 3.11, will remain. BACK UP YOUR SAVES by uploading them to online storage (PS+ required) or loading them to a USB drive.

I hope this helps!