They didn't say anything in the movie about investments, but Zaroff does mention the huge estate in the Crimea that he was raised on. I think it was made pretty clear in the movie that he had some serious money to fund his 'hobby'.
Movies I've Seen:
http://imdb.com/mymovies/list?l=25003655

The OP asked what they were living on. In the story, Zaroff explains that he came from a wealthy family and had his money in American investments. He had everything imported from the mainland. Maybe that's not made clear in the movie though. It seems that the movie is quite different from the story. I really want to see this!

Hmm . . . well, I wonder how big a deer park is, exactly? Is there a particular size it has to be under before it's no longer a deer park?
Here's a page about deer park hunting:
http://ezinearticles.com/?Deer-Park-Hunting-Basics&id=845788
It says, "This kind of a park is simply a state or county managed forest land that allows hunting in order to help keep the animal populations in check."
So, let's see, a state managed forest that allows deer hunting . . . without even searching to try to find a big one (but maybe it is oneI don't know), I found "Sproul State Forest" in Pennsylvania. That is 280,000 acres. It seems like that would be plenty big enough for everything shown in the film.

and how exactly did that gigantic waterfall complete with dangerous cliffs and caves enough to rival Niagara fit on an island the size of a deerpark