Invent a thing. You probably need about a paragraph of exposition explaining how it is very convenient for your characters that the "insert new storage system here" had just come into common use around the time the ship set out from port. A sealed, black-box type of recorder, designed robustly enough such that it can withstand the pressure, it's outer layers are some type of composite that doesn't react with salt water, and it's interior is filled with a non-reactive gas (argon perhaps) at some positive pressure, which will reduce the overall stress when the external pressure begins to climb. Information transmitted via wireless into the box, so there is no way for water to get in and damage the electronics. Battery life would be irrelevant, attach it to a power source once you get it back to the surface and crack it open.
There probably isn't a current thing that will do what you want without it being a bit contrived. A safe could possibly work, there are certainly steel containers which are strong enough and can be made watertight (plus the pressure will keep the door clamped shut), and could even survive centuries of corrosion if thick enough and relatively undisturbed - once a thick enough layer of corrosion forms, it may prevent further corrosion. But for this to work, your ship's captain would have to store all information in the safe, taking it out whenever he needs to use it, then immediately putting it back when finished. Not very realistic, nor is there a really compelling reason for that level of documentation and preservation aboard a boat.