Is it unsafe to hide a power strip in a cardboard box?
Its not particularly risky, but its not a fantastic idea either.

The amounts of current you are talking about pulling is not great, so heat dissipation should nor be an issue. 

No one will give you a guarantee that it wont catch fire, but it us highly unlikely provided none of the cables have exposed wiring (which might short and spark).

Why not use a plastic container instead? (Assuming a cable management solution as @HazardousGlitch mentioned does not do it for you.)