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Experimental evidence has been found for the existence of small single wall carbon nanotubes with diameters of 0.5 and 0.33 nm by high resolution transmission electron microscopy, and their mechanical stability was investigated using tight-binding molecular dynamics simulations. It is shown that, while the carbon tubes with diameters smaller than 0.4 nm are energetically less favorable than a graphene sheet, some of them are indeed mechanically stable at temperatures as high as 1100 C. The 0.33 nm carbon tube observed is likely a (4, 0) tube and is indeed part of a compound nanotube system that forms perhaps the smallest metal-semiconductor-metal tubular junction yet synthesized.