>Are we going to run out of space?

In a certain sense, we do. Pumping thousands satellites to LEO increases probability of triggering the Kessler syndrome. Luckily, LEO orbits are also self-cleaning on reasonable time scales (decades), so I think that some day we will trigger it (potentially, with some "help" from anti-satellite weapons) after which some kind of international regulation will be introduced to prevent repeating it in future.