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You don't run into a physical boundary, per se. Instead you'll probably run into floating point errors that start compounding as you get farther and farther from everything else. Kind of like how in Minecraft the world generator starts becoming more and more bugged out as you reach coordinates exceptionally far away from the map's origin point. Except in the case of KSP, the compounding errors usually mean that each part of your ship eventually flies apart and things go boom.
That's how things used to happen in older versions of KSP, at least. I'm not sure if the same really happens anymore since they revised the way universal coordinates work way back in version 0.17