Can I start over in No Man's Sky?

The only way I know of to start over is to delete your save file. That will reset the game. You could probably back up the save onto a USB storage device first, if you wanted, because there's no visible way to "switch" between saves.

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On the PS4, go into your settings, then go to the menu item labeled "Application Saved Data Management", and then delete the data for No Man's Sky to reset.

I found this: https://www.reddit.com/r/NoMansSkyTheGame/comments/4x5fjt/how_do_you_restart_the_game/

explode13 mentioned removing the following folder: NMS(or the one inside actually) under the path: %APPDATA%\HelloGames\NMS . If you cannot find folder AppData, just type it in the bar and the continuation will apear. This worked for me and I got to the tutorial from the beginning of the game.

Hope this helps.