Barber Paradox

Similar in style are:
• The astrologer who advises those astrologers who do not advise themselves
• The robot which repairs those robots that are non-self-repairing
• The catalog that lists those catalogs that do not list themselves
• Grelling's Paradox
• Berry's Paradox

The good point

The problem needs to be rephrased as such: A barber shaves all those men that do not shave themselves, and only those men who do not shave themselves. Otherwise, there is nothing wrong with him shaving himself. This also affects the self-repairing robot.

The bad point

There is nothing wrong with this problem if the barber is female.

The easiest way of phrasing the barber’s paradox so that it fulfils Russell's paradox is to say that there is a town five hundred miles from anywhere. In the town, full facial beards are illegal (but only for men, women are allowed to grow beards if they want to/feel the necessity, since that doesn’t screw up the paradox). There is only one barber, who is a man. Every man in the town falls into one of two sets. Each man either shaves himself, or is shaved by the barber. The sets are mutually exclusive. This is, as far as I can see, the closest we can get to actually applying the paradox, however the argument could be made that the sets are not exclusive, and that there is an intersect, the barber himself.