Q:

At what depth on Mars would the atmosphere have equal pressure of that on Earth?

I know the atmospheric pressure on Mars is less than that of Earth, and so is its gravity.  However, I know that the deeper you go (e.g. in a cave or a hole that is dug), the more atmospheric pressure you would experience.  How deep would you have to go on Mars (e.g. below the surface) for the atmospheric pressure to be equal to the pressure we experience on the surface of the earth?

A:

Nasa has a atmospheric model of mars: 
$$0.699 *e^{-0.00009 h} $$
A naive application of this model, solving for a pressure of 101 kPa, gives a depth of -55 km. 
The Armstrong limit depth (at which water boils at body temperature) is -24km
The model assumes constant temperature, and gravity (it doesn't correct for the fact that at 55 km below the surface you would be well into the martian mantle and the temperature would be very much higher, and deep enough for a measurable difference in gravity). There's no "goldilocks depth" at which you would only need an oxygen supply.
These depths are not achievable with current technology. The deepest mines on Earth are about 4km deep, and even the Kola superdeep borehole only managed 12km

