Containing a gas with pressure equivalent to the outside air requires just a membrane. Storing a vacuum requires a container that can resist external pressure of ~ 14 psi. It scales very badly.
This also creates an insane amount of stored energy in the stress on the shell. Here's a couple of videos of vacuum crushing steel tanker railroad cars [0] [1]. Of course that size doesn't even begin to scale up to the level of vacuum that would be required to buoyancy in the earth's atmosphere.
I'd think the only way to do it might be to have many small hollow spheres contained in a net or something, but the trick would be for each one to be light enough to be positively buoyant...
Yeah it's nifty idea -- no need for explosive hydrogen or scarce helium. If the polymer sheet is strong enough, I wonder if one could use it to wrap a sort of scaffolding (a shape like https://en.wikipedia.org/wiki/Fullerene but perhaps with struts through the interior for more strength), then evacuate the air from the interior in order to form a lighter-than-air "box". Submarines can withstand a few dozen atmospheres of pressure, this would have to withstand .. one, I suppose, so intuitively it seems plausible.
A helium balloon makes sense, it's lighter than air so it floats, duh.
Well, what's lighter than that? Nothing (vacuum)! It's such a dumb, but correct answer that I had a hard time wrapping my head around it.
Yup, theoretically a vacuum would be extremely buoyant if we could put it in a light enough structure.