No, California is not going to fall into the ocean. California is firmly planted in the upper part of the Earth's crust at a location where it spans two tectonic plates. But while the Big One would undoubtedly cause massive destruction, it wouldn't sink part of California into the ocean or separate the state from the rest of the country. The idea stems from a misunderstanding of the seismic forces that cause earthquakes in the region.
Powerful earthquakes frequently occur along the west coast of the United States because the region is close to a boundary between two tectonic plates. If you've read How Earthquakes Work, you'll know that the Earth's surface is made up of large, rigid plates that move slowly over the layer of the mantle below. At the boundaries between the plates, several things can happen. The Pacific plate and the North American plate simply smash against each other: one creeps slowly to the northwest and the other to the southeast.
Friction builds up along faults because the two sides come together very tightly. If the frictional force exceeds the forces that move the earth, the two sides will lock up and stop creeping. When this occurs, stress builds up along the fault line until the force of motion is large enough to overcome the frictional force. The pieces of land are then suddenly placed in place, releasing a large amount of energy that causes earthquakes in the Earth's crust.
Many scientists estimate that there is enough tension built up along some closed faults in California, so that when they finally slip, the earthquake will be extremely powerful. The Hayward Fault is of particular concern to these scientists because it extends below densely populated areas in and around Los Angeles. The idea that part of California will separate was probably inspired by the San Andres fault. After all, since the fault crosses California, one part of the state is on the Pacific plate and another on the North American plate.
If those plates move in different directions, it makes sense that the two parts of California also move in different directions. And this is, in fact, the case. But, even in a massive change along the fault, the plates travel an incredibly short distance, a matter of feet in the most extreme changes. The tension cannot increase to the point where an entire land mass moves many miles relative to another, so you won't see any significant piece of land separating from another.
Instead, pieces of land will move away from each other very slowly, taking millions of years to make large-scale changes. One end of California can slowly drift until it finally falls underwater, but it can hardly be interpreted as sinking into the ocean. Despite rumors that a major earthquake will cause California to break into the ocean, there's nothing to worry about. This is impossible to occur from an event in the San Andrés fault, due to its type of fault.
Without a functioning infrastructure, the local economy could easily collapse and people would leave Los Angeles.