Another great question sirjohn! We have hovercraft which travel on a cushion of air (http://www.hovercraft.org.uk/) and helicopters can hover if they need to. The trick is to design something with enough lift (force upwards) to balance out the urge to fall caused by gravity. If those two forces balance, a machine should be able to hover. I think we have the technology, we just haven’t been that interested in using it! Probably because both hovercraft and helicopters need quite a lot of parts to make them hover, and that takes up lots of space.
Although we don’t use hovering we have created levitating transport systems which is similar. These transport methods use magnets to lift the vehicle (normally a train) off the tracks and then propel them forward. The first commercial one was actually in Birmingham! There are a number of systems in development and some in operation reaching very high speeds.
In fact, inspiration for these hovering machines sometimes come from nature. Scientists have looked at imitating the flying motion of insects and birds to build unmanned flying machines which have uses in the military.