The way I see it, you have 4 options:
Free fall. As pointed out, you won't do this for very long before hitting the ground.
Go in one direction really fast (Orbit). This will result in extreme air resistance if you don't hit anything else, causing intense heat and probable incineration. The energy required to achieve this kind of acceleration would also extraordinary, so it would just be easier to orbit in space.
Go to the center of the earth (Where gravity pulls equally in all directions, so net zero). If you do this, you will promptly be crushed by the immense weight of everything else around you.
Compress the air around you so that it has about the same density you do. This is why swimming feels weightless. We are talking several orders of magnitude compression here, so in this case, you will be crushed almost instantly and your "weightless" crushed body will float in the special tank you built.
Not sure if any of the options are very appealing, but as far as I can tell, there aren't any others