That linked nature paper seems to indicate 10g/cm^2 for a 50% reduction. A standard shopping bag is about 5g, so you would need roughly 20k bags/m^2, or ~2k bags/ft^2. At 0.9g/cm^3, that would be roughly 11cm of solid polyethylene

I just squished a plastic bag as much as I could, and got it down to an inch cubed. So if I hypothetically did that for 2000 bags in 1ft^2 (sorry for English units), I think that would mean it could be 13-14in thick. So maybe it is reasonable to attenuate radiation by about 50% in a rather generous drop ceiling?

Ah, so bags are out, as is practically the ceiling. But cut sheet HDPE is easy to find, so tiles atop your machine would be relatively cheap and easy.