Is it possible for me to place my game into a dropbox for school?

Yes. Once you click "build," you can put the game files just about anywhere you want.

If you want someone to be able to play it online, though, without having to download it, the easiest way is to zip the files, and then upload the zipped folder to textadventures.co.uk. Go here to see that feature. When you are uploading the game, there's an option to make the game "unlisted" if you don't want to make it public. Then you can just give the link to whoever needs it, and whoever has the link will be able to access it.