The Long Dark Teatime of the Soul is that state which one's soul enters at about 3 o'clock on a Sunday afternoon, when one has had all the baths that one can usefully have that day and no matter how long one stares at an article in the paper one will never actually read it, nor use the revolutionary new pruning technique it describes.
       --Douglas Adams (paraphrased)

Update 2001/08/01: Ah, I see the esteemed flamingweasel has placed the correct quotation here. So I suppose I'd better actually justify the existence of my writeup here.

In all likelihood, the Long Dark Teatime of the Soul is actually a reference to an old medaeval belief known as the long dark night of the soul. It was, basically, a period of bad luck or other suffering which a person would be subjected to prior to some sort of divine experience in their life -- being spoken to by God, or what have you. Which fits with the plot of the book fairly well.