An Incomplete Death
Completely halting the aging process is not within the reach of human technology, and will (in my opinion) probably never be fully attainable. However, it is nearly within reach to bypass the aging process, by growing new body parts from your own stem cells and replacing the old parts with the new. Or, you could get a transplant of your brain into the body of someone else (after it is possible to reliably reconnect a severed spinal cord, possibly using stem cells.). This assumes you can find a brain-dead patient whose family is willing to donate his/her body to you. Finding a surgeon willing to perform this ethically questionable operation may be diificult as well. This may be repeated (as needed) indefinitely, or at least until something better becomes available.
Aging of the brain itself cannot, unfortunately, be bypassed (you cannot replace your brain). However, aging of the brain is a very slow process, and I have not heard of a case of death by brain failure. “It is really clear that if you don’t have a specific disease that causes loss of nerve cells, then most, if not all, of the neurons remain healthy until you die” [http://www.usc.edu/hsc/info/pr/hmm/01spring/brain.html]. But to prevent a loss of function in old age, a brain must be exercised (i.e. use it or lose it).
As technology accelerates in development, new methods for extension of life emerge. One can potentially ride one life extension to the next, and eventually extend his life until a permanent solution is found, such as digitizing the human mind. Some people choose to have themselves cryogenically preserved, to jump straight to a point in the future when they may be reanimated, and their lives may be extended much longer than possible in the near future. This preservation involves killing you, replacing your blood with antifreeze, and freezing you. There are no guarantees that the technology for reanimation from such a state will ever appear.
A simpler, and maybe cheaper, way to become “immortal” is to have your name and person remembered forever after death (not just as an entry in a database, though). Death becomes incomplete. Of course, doing something that noteworthy is very rare, and may not be open to everyone.
Within the realm of magic, an excellent example of “cheating death” presents itself: Voldemort and his Horcuxes. By keeping a part of his soul, and thus a link to the world, in several worldly items, he could not die. The Horcruxes had to be destroyed first. If he had not had enemies who did just that, he would in fact have lived forever.
In conclusion, one becomes immortal by being partly or completely bound to his or her realm, either through body, mind, soul, or legacy. A sacrifice must be made either way to do so.