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I just wanted to add to this that you actually CAN use an external HDD and play games off of it, but you will need to remove the OEM HDD, use a SATA extension cable to connect to the PS4's standard HDD port, and use an external enclosure for the HDD. Essentially, you are really just replacing the internal HDD, but using a longer cable to allow it to live outside of your PS4. This will allow you to use 3.5" drives, which gets you more drive space for the dollar.
I suggest just swapping it out instead, Sony makes it very easy, and you can swap out the factory HDD for a standard 2.5” hard drive, just like a laptop. I would just get a 1TB. You should be able to pick one up for $70-80. Check out the instructions in the link on how to upgrade the PS4 HDD, it is pretty simple and would only take you a few minutes. I did it with my PS3 and highly recommend the upgrade for any PS3 or PS4 owner that plays a lot of games.