The Earth is not a perfect sphere, so the distance to the center of the Earth varies from 6378 km (3963 miles) at the equator to 6357 km (3950 miles) at the poles. (NASA Helios Center). Challenger Deep, in the Marainas Trench got its name from the British survey ship Challenger II, which pinpointed the deep water off the Marianas Islands in 1951. Then in 1960, the US Navy sent the Trieste (a submersible - a mini-submarine designed to go to such extreme depths) down into the depths of the Marianas trench to see just how far they would go. They touched bottom at 35,813 ft/10,915m. That means, while they were parked on the bottom in the bathyscaphe, there were almost seven miles/11km of water over their heads!. (Extreme Science).
The problems in going any deeper, at least on any landmass, is the increasing pressure and temperatures. Workers in the deep gold mines in South Africa, which are at a of depth 12,300 feet, or 2.33 miles, experience temperatures of a blistering 70 degrees Celsius. As an example, working places take hours to reach, by not one hoist but a series of hoists at different depths. The technology does not exist to build a single-pull hoist to these depths. The weight of the rope itself would swiftly overcome its own strength, and it would break. (Business Week, 2004).
By the way, the deepest oil well is at a drilling depth of 31,441 feet...