With some rifles it matters more than others. A rimless case, it's more likely to cause issues, since the cartridge headspaces on the shoulder, and if the chamber's too long the case will have to stretch to fill, and could tear around the base and spill pressurized propellant gas back at you- a casehead separation.
A rimmed case headspaces on the rim, which provides an extra seal and means headspace matters much less. However, the Enfield has rear-locking bolt lugs and receivers known to stretch over time, meaning the bolt won't necessarily hold the rim flush with the chamber mouth and give a good seal.
With Mosins, where the bolt is front-locking and the casehead is rimmed AND enclosed by the rim of the bolt face, it's almost irrelevant, and in other rifles like the Steyr 95, it doesn't matter much either.
Also, Arisakas and Mausers, it's a good idea to verify CHAMBERING, not just headspace. 8mm-06 and 7.7mm-06 redone rifles are floating around out there.