Scotty: "A finicky transporter is nothing to gamble your life on captain."  And indeed it this is true!
So what would it take to build a transporter like the one's on Star Trek?  A lot!  A lot of technology that we simply don't have at the present time.  Actually, we do have the storage space it would take to map out your entire body, one atom at a time.  Since there are about 100,000,000,000,000,000,000,000,000  (100 septillion) atoms in your body, it would take a library full of CD's about 500 miles square to do it and it would take days, weeks or more accurately, millions of years to transfer the data onto the said storage medium but we could probably handle that problem if you could sit still for that long.  The next major step would be to scan all the atoms in your body.  To check their position, charge, orientation, quantum states, ect.  We can't do this yet, and even if we could, scanning with out changing what we scanned would be a major hurdle to overcome in itself.  Just simply opening your refrigerator to scan what you might have for dinner changes everything because the atoms of the things in your fridge have soaked up photons from the light bulb!

But if we could scan all the atoms and their properties in your body without changing them with the scan, we would then have to figure out a way to harmlessly disintegrate you, then transport all your atoms to another place.  This hardly seems likely the way it is done on Star Trek because we would need to have a machine at the other end of the transport to put you back together again.  You are reading this webpage after it has been transported to you, but you have a machine that puts it back together again.  Being able to transport things the way they do on Star Trek would be like you not only downloading this page, but also downloading an entire computer as well.

So you may wonder if anyone is working on teleportation seriously.  The answer is yes.  An issue of Scientific american a couple of years ago had and IBM ad stating that scientists at IBM had successfully teleported a "lump" of matter from one place to another.  So far though, no one can seem to dig up any specific information from IBM regarding this incredible feat.  There is a team of scientists at the University of Innsbruck who are trying to build a machine that they hope will teleport a single photon.  (A photon is a particle of light.)  If they can accomplish this feat they should be able to teleport electrons, protons and neutrons.  Once they can do that, teleporting an  entire atom will be only a few leaps away.   Marc  Millis, Aerospace Engineer, at the NASA Lewis Research Center, is currently the leader of a group of about 150 scientists and engineers that represents a program at NASA known as The Breakthrough Propulsion Physics Program.
Marc has this to say regarding teleportation:  "Transporters are only at the very first step of evolution. - conjecture.  We know what we’d like to accomplish, but we haven’t got a clue what physics we’ll even need to learn about. The closest topic to a transporter that we do have some understanding of is wormholes.  Because this subject is much less developed than the search for gravity control or hyperfast propulsion, there is little or no work being directly aimed at the question of transporters."