How deep can underground tunnels be within the scope of science?
If the Earth's interior was not hot and liquid, how deep could we dig before the tunnels collapsed by the pressure or the air became too pressurized to be breathable? In other words: how deep can underground tunnels be and still be scientifically plausible?
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1 answer
Problems for humans
The deepest tunnel-like structure created so far is the TauTona Mine in South Africa. The bottom of it is 2.4 miles below the surface. Temperatures there can reach 131°F, which has proven deadly many times in the past. The temperature can be lowered to more pleasant levels, but conditions are still very dangerous. Mponeg reaches similar, if not higher, temperatures (without cooling).
As Monty Wild mentioned in an answer to a different question, ventilation will be an issue for any underground structure. If you want people to survive this far underground, either they'll have to carry oxygen with them, sealed regions will have to be made, or many ventilation shafts will have to be dug, which will severely complicate the process.
Pressure is also a problem. We can use the barometric formulato calculate pressure (valid, I believe as per the derivation, e.g. here). It is $$P=P_b \times \left[ \frac{T_b}{T_b + L_b(h-h_b)} \right]^{\frac{g_0M}{RL_b}}$$ At, say, -15,000 meters (the missed goal of the Kola Borehole), this is $$P=101325 \times \left[ \frac{T_b}{T_b + L_b(-15,000-0)} \right]^{\frac{9.81 \times 0.0289644 }{8.31432 L_b}}$$ $L_b$ (the lapse rate) is calculated by $$L_b = \frac{dP}{dh} = - \frac{mg}{kT}P$$ You can plug this in, figure out $T_b$ from empirical data) and then solve for $P$ (after a bit of algebra). You won't like what you find.
Problems in general
Rocks are also under pressure. This is not good for excavating nor keeping what you've excavated intact. The tables can give you a good idea of what you're dealing with. A Powerpoint presentation downloadable here explains the pressure on tunnels and other structures.
The thing is, there's no good answer here. All of the really deep borehole were created for scientific drilling - experimental holes to see what's really down there. I can give you figures and equations and estimates aplenty, but I'd be lying if I claimed that I could give you an actual figure. We don't know what's down there. I gave you some necessary information to show you what you're up against.
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