Short answer: If you wanted to dig on Mars to reach a depth where the pressure would be 1 atmosphere, i.e. equivalent to sea level pressure on Earth, it would most likely be much too warm.
Long answer: Consider the case of Death Valley on
Earth. Since it lies below sea level, the atmospheric pressure there is
actually greater than what's found at sea level, roughly
1.01 atmospheres. Similarly, we could dig below the surface of Mars so
that the weight of the overlying atmosphere would be the equivalent of 1
We can calculate how deep a hole one must dig by using the "scale height" - this is the difference in altitude needed to produce a factor of e = 2.718x increase in pressure. In Mars' case, this is equal to 11.1 km.
Now, the pressure at the surface of Mars is a measly 0.006 atmospheres, while we want to go to 1 atmosphere. The number of scale heights we want to dig is then:
ln (1.0 / .006) = 5.12 scale heights
...which, for a 11.1 km scale height means we want to dig 5.12 * 11.1km = 56.8 km. Note that this is over 4 times deeper than the deepest hole ever dug on Earth, so this is already a pretty tough technological achievement.
Now, how warm would it be when we get there? For this, we need to consider the adiabatic lapse rate; this tells us how much the temperature drops as we ascend in the atmosphere, or similarly how much the temperature increases as we descend. (It's also for this reason that Death Valley has the highest temperatures recorded on Earth.)
In the case of Mars, the adiabatic lapse rate is 4.4K/km. In other words, for every kilometer we descend, the temperature increases by 4.4 K.
Thus by descending 56.8 km, we're increasing the temperature by 56.8 * 4.4 = 250K. Since Mars' average temperature is 223 K (= -50 C, -58 F), that means the final temperature at 1 atmosphere of pressure would be 473K (= 200 C, 391 F).
- This is unrelated to whether Mars has a "dead core" or not. This temperature increase is not due to geothermal (or in this case, areothermal) energy. Rather, it's a simple consequence of taking the current atmosphere and compressing it adiabatically as it fills up our hole. A similar transformation would be suddenly opening the doors on a pressurized jet at 33,000 feet...the air would quickly expand to the thin ambient pressure and cool down in the process by 65o - 98o C, depending on how humid the air inside the airplane was.
- You can't generate electricity from this temperature change. It seems counter-intuitive, but even though the temperature has increased, there's no extra energy added to the system - this is the definition of an adiabatic transformation