In reply to Terminal_Velocity:
I have eczema (as well as all the other typical atopic problems: asthma, allergic rhinitis, sun-activated dermatitis etc.) and have had problems with chalk from time to time. This tends to be mostly a problem indoors (I guess as my fingers sweat more indoors and so I use more chalk). It was worst when I was using Metolius Super Chalk as it has a drying agent, so I would recommend you not using that.
Aside from that, wash your hands at the wall when you are finished climbing. When you get back home wash again and use a dermatological moisturiser (E45 or similar).
No idea whether it is coincidental or not, but seem to have less problems since moving to using loose chalk rather than chalk balls.