Technically levitation is completely possible. Even more so, there are at least two ways to achieve it.
Let me describe the principle first, so that you will see what I mean. Levitation is like floatation, just that it happens in air. In water gravity is countered by bouyancy, this being the Archimedes law, found by its name bearer while bathing and then jumping out of the bath and running naked in the streets and shouting Heureka!
Buoyancy in air is not enough to counter gravity so a different force is needed. There are several options, of which force by air thrust is just one. Helicopter and harrier fighter airplane both use this principle of levitation.
Another extremely interesting option is the electric field of the Earth. Few people know that the natural field around us is 100 Volts per meter. Any persons head is almost 200 V higher than his or he feet. Amazing, right?
If we could tap into this field by opposing it this could raise forces of magnitude easily comparable to gravity. A 100 kg heavy object could levitate if we could uniformly control 10^20 of its electrons. Seems like a lot, until I tell you that a typical 100 kg object has roughly 10^40 electrons altogether. So, all we need is to get about one in a hundred billion billions to comply to our ordering.
Aha, we need an army to march to our tune. How can that be done? One idea: resonance. Another one: spinning tops with one prefered spatial dimension.
Resonance is hard to pin down but it does have to deal with specific quantum ordering of living matter. Hereby we see that living organisms are a seriously advanced technology that we will not be able to master in our existing scientific paradigm alone. Much more is needed to even alow ourselves to think differently.
Spinning tops are easier to explain. If we can get permanent magnets to spin in such a configuration, that the resulting electric field exerts a counter force to gravity this directly results in levitation. The configuration and spinning should be achieved in stable conditions with zero or no friction. I imagine that this could be done in nanoscale, elswhere friction would kill the experiment immediately.
All of the above, naturally is a provocation, yet it is meant to rethink some of the old and uncontested truths. Maybe, just maybe by thinking differently we could one day indid achieve levitation.