I built it out of some kind of foamy board I came across at a craft supply store, it's very light, 1/4 inch thick and can easily be cut with a knife and glued or taped together. The little 'bars' that hold the boards apart are also cut from the same foamboard, wrapped with tape to make sure they would not leave any 'crumbles' on the paintings.
As you can see the box does have a lid, but I haven't decided on a way to secure it closed - I might just use some rubber band, that'll do nicely.
The finished box actually feels more sturdy and robust than I expected. I might even cover it with adhesive foil to make sure it doesn't melt if it gets wet, but then again it's small enough to just put in a plastic bag if needed.
The box will hold ten boards, so I might have to make a second one later on - I must say I'm quite proud of the little thing, though there's still room for improvements ... ;-)