Heat based limit
The TauTona Mine is the deepest mine on the Earth. It is a gold mine.
Tip of the TauTona mine iceberg:
- Depth - 2.4 miles
- Ambient Air Temperature - 55C (131 F)
- Rock Face Temperature - 60C (140 F)
If the air conditioning stops working, the ambient air temperature can kill the miners. On average 5 miners die every year in this mine.
I assume that without active cooling, this is below the lowest inhabitable depth.
Less than 2 miles of depth
Oxygen/Carbon Dioxide Based Limit
Underground there's very limited life able to convert our wastes ($CO_2$, urine, & feces) back into forms that we can use. Although we have many mines located a mile or more below ground, all of these require air exchange to keep humans alive.
Very deep & active mines require the use of active system to exchange air with the surface (e.g. fans & blowers) to keep the miners from suffocating. If your blowers stopped working for more than an hour or so, expect your population to begin suffocating.
If you assume an active air handling system that never suffers failures, then keeping the air fresh is:
Not a limiting factor
It should be possible to create a passive air exchange system using the dynamic pressure of wind passing over special chimneys and other techniques to exchange air with the surface, this probably limits the depth of the mine too.
Without actually modeling such a system, I don't really know what the limit would be. One unfortunate fact is that $CO_2$ gas is heavier than $O_2$, so it will tend to settle to the bottom of your hole.
If you assume a passive air handling system, then you're limited by the volume and forces such passive system can generate. The exact answer will depend strongly upon the details of the configuration so I can't give you an exact answer.
Unknown, but I would guess less than 1 mile
Closed Loop Life Support
If you built your secret place more like a permanent space colony which only needed the input of energy to keep it going, then this wouldn't be an issue. However, it does point back to the first part of the answer in which you still need to find a heat sink to keep the place livable.
Getting energy isn't a problem, however, dumping your waste heat will become a problem at depth. My guess is the limits of your ability to dump waste heat will require you to limit your maximum depth even more than the open-loop environmental systems would.
A closed-loop life support system would ensure that you would not need to place your secret hidey hole near the surface to maintain a supply of fresh air.
My guess is 1 mile < your maximum depth < 2 miles