In a reply to another comment I gave a link to the type of trash can that I was thinking about for the cage.

While there may be imperfections in flatness on standard, uncarpeted floors that can allow RF signal to find a path inside (reflections from below or from other angles as another comment mentions) or you could live in a house like mine where flat had a non-standard definition with lots of slop, I believe that you solve those issues by setting the lid upside down on the floor, placing your apparatus inside the lid, then covering and sealing it with the inverted trash can.

You should have plenty of signal attenuation either way.