Or, Make a clamp of some kind to hold a sort of short mobile antenna on the balcony rail and add your own "ground" or counterpoise. Call it a 'counterpoise' since that's closer to what it actually is, the 'other half' of the antenna. Or the whole thing a 'single radial' antenna. That counterpoise is just a wire about 8 - 9 feet long attached to the braid of the feed line at the base of the antenna. That wire can be dropped over the hand rail of the balcony, laid out on the floor, whatever, just don't coil it up.
You tune that sort of thing just like you would any other antenna, by adjusting it's length. It takes a while sometimes but it certainly will work. The 'best' in the world? Uh, no. But certainly better than no antenna at all.
- 'Doc