Many of us who grew up in the United States learned an expression about digging to China -- the idea being that if we could dig a really deep hole, we would come out on the other side of the Earth, and that would put us in China. I still say this to my geography students quite often, testing whether they are awake and paying attention.
Assuming super-powerful digging equipment and super-human digging crews (not susceptible to huge pressures and high temperatures), we could dig from the United States to China, but we would have to begin our hole at an angle of about 45 degrees in the right direction, not straight down.
If we were to dig straight down, where would we end up? At our antipode, of course! This article explains that since the earth is covered almost 70 percent by water, most people who start on land would end up coming out in the water on the other side of the earth. The article also explains how to figure out exactly where that would be and describes a few of the more interesting pairings -- including the people who really would be able to dig straight through to China.
The article also points to a very interesting site -- Antipodes Map -- that enables users to view antipodes of any place on earth on paired Google maps.