Generally, you'd want to check if it's on a streaming service you already have access to. That's the fastest way.

Then, check your library.

If both of those fail, you run into choosing to hunt a copy down, or sail the seven seas.

Stuff like Kurosawa has a big enough fan base that you're very likely to be able to find used copies fairly cheap. The more obscure the movie, the harder it will be to find, and prices can vary depending on how much of a cult fan base it has. Sadly, eBay really is the best option in that regard. So you don't always have to pony up for the more expensive editions.

Since you don't like accumulating plastic, you can always resell, or donate to your library, depending on how willing you are to lose money on the swap.

You can check local message boards too. Sometimes folks will give stuff like that away. Pawn shops still occasionally have DVDs around. If you can find film buffs in your area and make friends with them, you may be able to borrow stuff, or have a movie night. Those of us with big collections do tend to like hosting movie nights to share the fun, though you have to rely on making connections with people that happen to have stuff you're interested in.

And, yeah, there's file sharing options as well.