This is a fun what-if. I really wish I was xkcd right now so that I could include pictures and humour, but I'm not.
If we built a giant ring around Earth at just the right height everywhere, then in theory, the ring would just float there once the supports were removed. However, (and this is the reason not to try it) this is an unstable state. If you so much as tap on one side of the ring, the whole thing will come crashing down. It's like balancing a pencil on its point or a bicycle upright while stopped (no training wheels!). Possible, but instability is a b!%#.
But let's back up to how you might go about doing this. The pressures on the ring would be enormous, so you would have to make it very thick and out of really strong materials. Carbon nanotubes would be great but expensive. Odds are we'd go with steel for cost and just make it thicker.
Then you need to build it in exactly the right place and at the right height. And the supports would be none too stable also (why would we not just build this as a giant global bypass and leave the supports in place?). But let's say you got it built. To pull off this feat, you need to remove the supports simultaneously and quickly. That won't be easy since they'll be huge and firmly in place. Best synchronized demolition ever!
Ok, now you have this ring floating. It had to be built above the atmosphere because any winds or atmospheric effects could bring it down. But also, it had to be built at the right altitude to avoid all that space junk we have floating in orbit. (One stray satellite and BAM! we'll bring a whole new meaning to the phrase "Hey look at that huge metal ring that's crashing down to Earth!")
And, to make this easier for us, somehow we got it spinning (not very fast) so that angular momentum and orbital mechanics could help us out (Okay, those don't actually help the ring, but neither do they hurt it and they make it look cool; a spinning ring in space. I call that a net gain, so it helps).
Before the day is out, we have a problem. We forgot one crucial thing. That darn Moon! The tidal forces from the Moon (wherever it is at that time) upset the delicately balanced and unstable ring system. Now we have something that we should have considered; the extremely massive ring of steel we built is crashing towards Earth. It's going to leave a dent and kill lots of folks. On the plus side, building a massive ring like that will look good on your resume.