I was wondering the same.
Would be funny, but I don't think barbs can take capital (assuming Great Wall would be built in Capital).
I had a screenshot of barbs knocking down one of Siam's cities (new, 1 pop), and it was reduced to 1hp, but they never took it. Barbs started chasing other units, including my scout which was about 6-7 tiles away from them. Will post it later.
Dunno, maybe barbs are scripted only to bug AI, but not to hurt him much, like taking his city. Never happen to me either, sometimes they will attack city, take around 3/4 of health, but then start chasing something else, like scout for some reason, which makes me think they are scripted only to pillage title and annoy in general. In early games, their pillaging is lot more troublesome then AI rush attack.
and I've seen often that AI runs around with his settler freely and no barb even capture it (rarely). but if my scout or missionary walks five titles away, of course they chase him around half the world.