Rather than a polystyrene ball, something like Isokinetics Inc.™ Anti-Burst Exercise Ball - Isokinetics Inc. might be a good base to make a globe. Add a little papier mache to smooth things out and then start drawing.

Or my personal favorite, a thrift store globe and a can of flat white paint. A little sanding would be needed to take out the earth terrain is present, but that's simple enough.

The topic of making physical globes come up once in a while.