How do airliners cross the ocean without GPS?
Well, most airlines do cross the ocean with GPS in today's world.

That being said, most (if not all) transcontinental airliners, and many flying domestic routes as well, have what's called an inertial navigation system (a form of dead-reckoning where gyros and accelerometers are used to compute changes in position). The INS feeds into the flight management computer which is used to cross-check the GPS position, and can be used as a primary navigation source should the GPS signal be lost or corrupted.

Prior to GPS, inertial navigation was the primary means of navigating across oceans. However, a great many airplanes did cross oceans before INS... with varying success. The general idea was to use a combination of dead-reckoning, radio beacons where you can find them (small islands and coastlines), the sun (with the help of a sextant) during the day, and the stars at night. Hopefully, this puts you in position after some period of time to pickup ground-based navigational aids, or to identify a coastline. This actually works pretty well if your wind estimates are good, and the margin for error is high enough. As DeltaLima points out, long-range radio navigation systems also started to emerge around World War II (LORAN and Decca) which had a much greater range than previous radio beacons, and therefore decreased the amount of time required to fly by dead-reckoning alone. By the time inertial navigation became prevalent in the 1970's, airlines were already providing reliable transcontinental passenger flights every day.

Another possible outcome is that your plane is lost in the ocean, never found, and the Discovery Channel funds an expedition to look for you 75 years later.