Can you own PS4 games?

Update: I am EXTREMELY OLDSCHOOL: my last gaming console was Sega Genesis. In between Genesis and Now, I have played a bunch of computer games, but only one game in the past 12 years or so.

I was thinking about buying a PS4.
Update 2: But, since I am old school AND cheap, I want to buy a game and then just own it and be able to play it as much as I want and for it to be available for the next 50 years. Is that possible? Or is everything purchased on a monthly subscription nowadays?
Update 3: Second, I really like first person shooter types with a good storyline. Do games like this still exist where you can just play on your own and eventually "beat the game"? Or is everything in this game type always multiplayer now with teams and no ending?

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