How does this system filter radioactive air?
The air is not radioactive, just the particles in it.
You don't even need a filter, just a baffle system so the particles settle out.
If you want to get fancy, you can use everything from dampened cloth to furnace filters or HEPA filters.
Better to just have a baffle system that you can avoid, because with filters you will now have concentrated particles in the filter to dispose of.