Some people swear by Altoids tins. I use a tin that contained tea, it's about 2 by 3 inches at the base by 3 inches tall. Think small. You can get a LOT of cloth scraps into a tin even without forcing it. If you're going to try and use some kind of like a cookie tin, broad and flat, the challenge is going to be heating it evenly so that you don't end up with a wide variety of different levels of charring. Something like a 2 gallon popcorn tin is just total overkill. A normal steel soup can would probably work fine if you have a side-cutting can opener and use mechanic's wire to hold it closed.

The thickness of the container isn't really a factor, the steel isn't going to get hot enough to have any problems just laying in an open fire. But solder would melt, that's a good point.