What kind of "emergency" would we be talking about here?
I don't know. Using my imagination, I can envision a situation where I have a rifle chambered in .358 at my disposal in a time of need (grizzlies or zombies at my door, whatever) but no ammunition for it. However, I have some .308 cartridges at hand. Could I shoot the rifle safely
is the salient question.
This question was not intended to be anything more than expressing a curiosity as to what one might expect to happen if a .308 cartridge was fired in a rifle chambered for the .358 Winchester. Nothing more than that.