"Dark of the Moon"

Dark in this phrase is used as a noun in the sense of "night." Consider that if we could see the moon at this time, we would be looking at its night side. The "dark of the moon" is nighttime on the moon.

While you could say the darkness of the moon, that would imply that the moon was causing darkness, which I suppose it does during a solar eclipse.

The examples you give in your comment—"Soft of the pillow," "Hard of the rock," "White of the fence"—sound funny because soft, hard, and white are all adjectives rather than nouns, so they don't fit this pattern. It would be odd to say the pillow's soft, the rock's hard, the fence's white, because we are waiting for the noun that those words describe: the pillow's soft caress, the rock's hard surface, the fence's white picket.

More appropriate comparisons would be "The Night of the Were-Rabbit," "Dawn of the Dead," or "Dark of the Sun." These are all "grammatical" phrases. They could equally be expressed as the were-rabbit's night, the dead's dawn, the sun's dark, although they sound less nice that way.

It's an usual expression/image, and so needs a bit more interpretation than other phrases. But there's nothing ungrammatical about it per se. If you can talk about something being "the dark", and you can talk about "the X of the moon", then it's plausible to talk about "the dark of the moon".

Consider also that part of the idea of the expression is a play on words, contrasting with e.g. "(by) the light of the moon", which would be a more common expression.

Or put another way: it's no less grammatical than "the cold of the staircase" or "the dry of the waterwheel".