I've got a nice Minecraft world on my friend's PS4, but I want to continue with the same world on my own system. Am I able to copy it and if so, how do I do that?
Minecraft for PlayStation consoles isn't directly developed by Mojang, the ports were taken care of by 4J Studios. As part of this distinction they will refer you to PlayStation support for something like this.
For non-PlayStation platforms this official help site explains how to do this using a free trial for Minecraft Realms, which feels like a weird workaround. But last I checked Minecraft Realms wasn't available for the PS4 version to begin with.
Slightly more promising is this article that explains how to migrate your PS3 Minecraft save to a PS4/Vita, that might be able to help you: (click)
I've also added the listed steps below:
While on your PlayStation 3:
- Start Minecraft and press Play Game
- In the list of your saved worlds, scroll down to the world you want to transfer
- Select a world and press R1 to open the save options
- Choose the alternative Upload save for PS Vita/PS4
- Wait for the upload to be completed and press OK when you’re given the option
When you've performed those steps on your PlayStation 3, start your PlayStation 4 or PlayStation Vita and do the following:
- Open Minecraft and click Play Game
- When in the game menu press the square to download your uploaded world
- Wait for the download to complete and click OK
You obviously don't want to transfer your world from a PS3 but instead from a PS4 to a different PS4, but it still might be worth looking around to see if this export option is available.