I've seen people saying that 5.8mm is designed to be compatible with 5.56mm in emergency, I am almost 100% certain that is a complete fabrication that originates from 地摊文学 (extremely unreliable sources). But since it's been around for so long you'd find a lot of sources referring to the same thing. Let me just say this, just because you see a lot of sources saying the same thing does not necessarily make it correct.
It is entirely possible to fire a 5.56mm bullet from a 5.8mm gun. The bullet will probably fit in and if you can close the bolt on the bullet and let the firing pin hit the primer, then it will ignite. But, this does not mean it is a viable thing to do. First, because the diameter of the bullet and the chamber does not match and will not create a perfect seal, the powder will not burn completely and your bullet will not go very far in an accurate manner, even if it shoots (you'll shoot most of the powder along with the bullet tip). Furthermore, the gun will not cycle properly and if you want to fire 5.56mm ammo in 5.8mm gun, it will be a bolt gun at best, and it will not fire every time. The final problem is that when the chamber is not sealed completely, the gas created by the burnt powder may not all be used to push the bullet head forward, it may exit from the weakest point in the casing and erupt the casing, making it stuck in the chamber, which can be extremely difficult to extract.
How do I know the above? From personal experience shooting a 5.56mm ammo in a 7.62mm gun. It was a stupid accident but it gave me some interesting insights on how this type of thing might work.
Now, if we are seriously talking about the viability of shooting 5.56 in 5.8 gun, you can probably do it and it will probably shoot. But, your gun will not cycle, and you may have to deal with erupted casing each time you fire. Furthermore, your bullet will not really go far since most of the powder will not be ignited. If the load in 5.56 ammo is hotter than what your 5.8mm chamber can handle, your gun will blow up in your face and shrapnel created in that may kill you.
So let me just say that 5.8mm rifles are not designed to be able to fire a 5.56 ammo under emergency situations.
Even guns that fire the same caliber like different ammo. 5.56 guns with different twist rates in their rifling perform significantly differently depending on the type of 5.56 ammo you fire, some like 55 grain tips, others like 62 grain tips. My G36's zeroing shifts by 3 inches above at 100 meter when I switch the brand of 5.56 ammo I fire. If you fire a 5.56mm bullet in a 5.8mm, it may fire, but it is not going to function in any meaningful manner.