I don't plan on making a chest of drawers but am interested in the techniques involved so I could apply them to other furniture.
EDIT: close up picture
It appears that those are thin pieces of tin nailed to a wooden cabinet with brass tacks. I think it is quite cool actually, sort of steampunk.
To do it large scale you would be best served with a sheet metal shear (think of a heavy duty paper shear), a pinch roller (adjustable twin steel rollers mounted vertically which you spin with a handle whilst passing sheet metal between, to set a curved bend), and maybe a sheet metal "brake" (a sheet metal bender for straight bends as opposed to curves). Also various metal working tools: vise grips, awl, tin snips (straight, right-hand, left-hand), various files, ball-pene hammers, small anvil, etc.
You can buy some "DIY" sheet metal work contraptions that combine a roller, shear, and brake for fairly cheap. Check on e-bay.