The fourth dimension
OK this might sound a little far-fetched, but since we're talking about floating objects...
All the objects in the world are four dimensional, and are bound to the ground in an area outside of our 3-D universe.
Skip to "to here" if you're familiar with the concept of 4 dimensions
To see why this works, you need to wrap your 3-D suited mind around the concept of 4-D space. (OK I know people say time is the 4th dimension but I'll ignore that for this answer).
Imagine if you will, "Flatland", a two-dimensional universe, containing only 2-dimensional people objects and people, everything in the universe lies on a single plane of existence, with just an X and Y axis. The flatlanders only have a concept of depth and breadth (or height if you wish), and they would find it impossible to imagine the concept of 3 dimensions.
Now imagine a sphere passes through their plane. This sphere appears to them as a circle, increasing, then decreasing in size as it passes, because they can only see the single slice of the sphere that currently intersects with their plane.
Now imagine a normal 3-D bridge, which crosses through their plane. They can't see the parts of the bridge connected to the ground, they can only see the ground, and... a floating platform!
Now we return to our world, we all can see in 3-D and only understand the concept of 3-D objects, but there is actually a 4th dimension and the objects are four dimensional, but they look 3-D to us because, like the Flatlanders, we can only see the "3-D" slice of them.
to here
But that floating object is actually connected to the ground somewhere else along the fourth dimension. It looks like a floating platform, but it's really a 4-D bridge or something, just fixed to the ground somewhere else we can't see.
Likewise (if we're to use Minecraft as an example) when you chop a tree, it doesn't fall down, because it's a 4-D tree. If you can imagine flatland vertically intersecting a real-life 3-D tree, they could "chop" the trunk and it would look to them like there is a gap with a tree floating above it, but actually all they've done is cut a vertical slit in the trunk so the tree is still perfectly capable of standing.
This has the advantage of explaining why floating objects don't bob or move about as you'd expect if they were actually floating in the air.
Finally, why do some objects fall? Well the easy answer is you could say not all objects are 4 dimensional, but you can also take a look at the substances that do fall: snow, sand - fluid substances that would flow down even if you cut a slit in it. You can't build a bridge out of sand!