They were talking about lead foil. Very thin sheets of lead. (Like aluminum foil, but with lead.)
Lead, on the whole, is very soft.
Obviously, the thicker the sheet, the stronger it'll be. But heavier, too. (Same goes for aluminum foil. Generic foil is thinner than the brand name stuff, and will tear much more easily. The "heavy duty" brand name foil is even thicker, and thus more tear resistant. But it's also bulkier, heavier, etc. You get less of it in the same size box, and it can be harder to bend and shape.)
Lead foil isn't exactly light. They mentioned on the show that it was something like ten times the weight of the aluminum foil they were using. And they had to work to find a balance in the foil they were using... thick enough so that it wouldn't fall apart or tear too easily (which it was still doing), but thin enough so that it wouldn't be too heavy to lift. (And even then, they had to make the balloon pretty big before it would actually take off. Consider how small the aluminum one was, and how well it went up and stayed up.)
But... yeah. It's possible to get lead foil in a variety of thicknesses. And if you're sewing it into a fabric as a lining, you don't have to worry quite so much about tears. It'd be heavy for a robe, but, depending on the thickness of the foil (and how much of the robe you wanted to cover), it could still be quite wearable.
