I need two halves of a hollow sphere, preferably made out of acrylic. Clearly I could take something round and cast it, but how could I make this hollow?
ShadoWorx has the slush mold right, but depending on your size, it might be easier to take ttdriver's suggestion. What you want are "clear ball ornaments" probably by Darice inc.
They come in many sizes. Honestly, the only reason I know the name is that I have one in front of me for a current project. Found em at a local michaels, though not every michaels carries them. Saw them in 2.75 inches and 3.15 inches. They have other sizes.
As for the mold, another post (http://www.therpf.com/f9/starcraft-ii-agr-14-c-14-rifle-210659/?highlight=cast+no+seams+rifle) member lightz describe how me made a two part mold, filled each half with a bit of resin and slushed it a bit. Then he filled the mold with more resin, sealed it, and slushed further to get the seam sealed. This produced a "seamless" cast that was still hollow but mostly thick in many parts. This would probably be what you are looking for to get a good sphere shape that is still hollow. Might want to shoot him a PM to see how successful it was though.
Acrylic hemispheres are available off the shelf from plastic fabrication companies in Europe and the States.
You just need to do a search for acrylic / plexiglass hemispheres in your locality.
What size do you need?