How does a submarine survive the pressure in the deep sea?

Submarine hull shape and its construction is designed specifically to operate at deep ocean pressures. Specifically, spherical shapes are stronger under extreme pressure, and most submarines are constructed of flexible metals that allow them to contract and expand as the pressure changes.

Of course there's a limit as to how much pressure it can take, and that also is determined by the vessel's construction and design. For example, the Bathyscaphe Trieste has been to the bottom of the Challenger Deep in the Marianas Trench, the deepest point known in the ocean; her design is highly spherical and small to withstand such pressures. By comparison, a Russian ALFA, with a Titanium hull, is rated at 3,000 feet crush depth - but it can't maintain it for long, nor can it go to such depths that often because Titanium is a brittle metal when repeatedly exposed to high pressures.