How deep would a hole into the moon need to be to provide air pressure suitable for human life? Say we dig the hole then fill it with air. How deep would it need to be? What equations would apply? Would this be a plausible SF scenario?

Your question is Very difficult, without making a lot of approximations.

I made a Very rough computation, with the following approximations:

1. Gravity doesn't vary along the hole and it's equal to (1/6)*g.

2. The air's temperature is uniformly at 25°C (300K).

3. You are satisfied if the pressure at the bottom of the hole is 90% of 1 atm (or the hole's height would be infinite, as it would be in the case of Earth's atmosphere thickness, with this simple computation).

Then, taking (1/6)*g = (1/6)*10 ms

^{-2} as the uniform gravity inside the hole, and using the exponential law of the pressure with height (barometric formula):

p(h) = p

_{0}*exp(-αgh)

p

_{0} = 1 atm

α = 1.2*10

^{-5} m

^{-2}s

^{2}g = 10 ms

^{-2}from: 0.9 atm = 1 atm*exp(-αgh)

I get: h ≈ 1.2*10

^{5} m =

120 km.

You can understand how rough are these approximations, the worst is probably the air's temperature...