The only way to experience the effects of zero gravity, without going into outer space, is to ride on a zero-gravity flight in an airplane. As the plane passes over the top of a parabolic arc, the cabin becomes a sort of zero-gravity chamber, during which time passengers experience near-weightlessness.
The term "zero gravity" is something of a misnomer. The plane, and everything in it, are still subject to the Earth's gravitational pull. However, as the plane passes over the top of the arc, the centrifugal force temporarily balances out most of the downward pull of gravity, to the point where the effect of gravity feels negligible. What passengers experience is more accurately called "microgravity."
The effect of near-weightlessness generally lasts about 20 to 30 seconds at a time, after which the plane completes its dive and begins to ascend again. As the plane pulls out of its dive and begins to climb, the g-force increases, and passengers go from weightlessness to feeling as though their weight has nearly doubled. These flights can cause extreme motion sickness, earning them the nickname "vomit comets."
Flights of this sort, long used by NASA to prepare astronauts for the sensation of weightlessness, have become available to the general public in recent years. Since 2007, a company called Zero Gravity Corporation has taken passengers up in a modified Boeing 727 and allowed them to experience microgravity. Generally, the pilot performs the ascent and descent pattern no more than 15 times, as a way of limiting the motion sickness some passengers experience.