We have used a 3-GHz microwave host cavity to study the remarkable electronic properties of metallic, single-walled carbon nanotubes. Powder samples are placed in its magnetic field antinode, which induces microwave currents without the need for electrical contacts. Samples are shown to screen effectively the microwave magnetic field, implying an extremely low value of sheet resistance (< 10 μΩ) within the graphene sheets making up the curved nanotube walls. Associated microwave losses are large due to the large surface area, and also point to a similar, very small value of sheet resistance due to the inherent ballistic electron transport.