This [1] is a pretty good description of the spider silk stucture. Turns out it's a self-assembled liquid crystal composed of oriented rods of a hydrophobic protein (an alanine/glycine co-polymer for those who care). It's not hard to imagine how a carbon nanotube could substitute for one of those if it's around in the soup...

[1] http://www.chm.bris.ac.uk/motm/spider/page3.htm