>cough<
Actually, parts of Chicago's Deep Tunnel network used for storm water management do exceed a depth equivalent to more than 20 stories.... but that's because they've utilized the Thornton Quarry as a reservoir. Parts of Thornton Quarry are 125m deep, roughly equivalent to a 40 story building.
I'll just throw in for free that in 2008 the area received such a quantity of water that the Deep Tunnel system
filled up entirely, including the whole of the Thornton Quarry. That's 3.1 billion gallons of holding capacity, or 11,734,776,530 liters (more or less - I'm assuming the gallon quantity involved rounding). Here's a picture of it:

(yes, that's a freeway going through the quarry, on a bridge). There's a new water run-off reservoir being constructed, scheduled to open on 2014, that will have about three times the capacity of Thornton, as even that massive capacity is considered inadequate for the region's flood control needs.
It is certainly within our abilities to build such structures. It's a question of whether or not we want to do so badly enough.