When I try to place more than 1 old safe on a map, it says "You already have an old safe", and won't let me build.
Did the team make this a design choice ? Or is it a coding oversight ? I can't think of any reason why a person could not have more than 1 safe in real life ?

Its by design for balancing reasons. Only 1 safe per person. It would be nice to let server admins choose how many a person can have, especially for pve servers where there isnt any pvp balancing to worry about.

Yes, I guess on a PvP server, players would just place safes all over the island and be able to hide away from the combat, grab another weapon and it would be a rinse and repeat episode.

Origins gives the impression of allowing complete freedom in choosing your style of play, but then design-decisions like this, where you are forced to adhere to someone else's idea of how you SHOULD play, kind of breaks the immersion. Origins should be fun too, not just a relentless grind of get illness from zed, commit suicide, run to body, repeat. But I do see the devs POV.

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