Headspace is an issue for reloaders, as the case will stretch and fire form to fill the void of the chamber, as it is supposed to do. These military rifles had generous clearances (read sloppy) between case and chamber wall to allow many different manufacturers' cartridges covered with muck, sand, blood whatever, to chamber and fire in wartime conditions.
The big issue these days is that commercial cartridges are not as robust as the mil spec stuff and reloaders squeeze the brass back to original size in the reloading dies. Do this a few times and the metal of the cartridge will fatigue from being restretched and resqueezed. This was never an issue with milspec ammo as it was literaly a one shot cartridge that was never intended to be reloaded.
Headspace is another issue for target shooters who try to eek the last bit of acuracey out of a round. Has to do with the 'set up' of the bullet as it is presented to the rifling in the throat when the chamber is loaded.
If the bolt has the same serial number as to that of the rifle, chances are that the bolt was matched up and is still good. However, if the bolt number does not match, it has been swopped at some time and there is no sure way to know if the headspacing is good unless checked. There are special guages which resemble a stubby cartridge with rim. These are precision machined to fine tolerances. Basicly you slip one into the chamber and gently try to close the bolt. If the headspace is within limits, the bolt won't close, if too sloppy it will. There is a bit more to it, but that is the general concept. There are a series of 'go' and 'no go' guages and are quite expensive, not something that a person would likely buy to use once, but are good to have if you collect and shoot these things.
There is also a fiddely way of using an empty cartridge and feeler guages to measure the headspace, but unless you know what you are doing, take it to a smith set up to do the checking.
I usualy measure and if I think it is ok, I will take the chance and fire the rifle and inspect the fired case for any signs of problems. But then again, I'm a guy that runs with scissors, taught myself how to blast with dynamite and made my own parasail rig.
Modern North American commercial specs call for a headspace max of 64 thou, original milspec calls for 74 thou and wartime emergency spec was 84 thou. 84 thou is rattling loose so that shows how much leeway there is with the specs.