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You can of course make the doors out of just about anything. If this project is expected to last however "big box store" wood, either mahogany or birch will not last too long. In fact it could fail in only a season or two. Real marine plywood has thicker surface veneers, higher quality interior veneers and interior veneers with no voids.
If you let us know where you are from I suspect some regular members will be able to help you with sourcing.
Also the design is an issue so you will get a better answer with pictures.