We aren't saying Earth would lose all its animals if sharks went extinct. We would, however, most likely lose some, and gain oversized populations of other species. To understand this, let me give you more context.
Photo by Cassie Paumard, Lead Marine Biologist/Founder of Project Kolika.
Natural selection is the theory that states those organisms best adapted to their environment have a better chance of surviving and reproducing. For example, there is a large population of sea lions feeding out at sea. Their predator, a white shark, appears. The slower individuals of the sea lion population are swimming at the back of the group. Because sharks go after the sick, weak (in this case slow), and dying individuals, the great white will go after the slower individuals, since they are an easy target. These slow individuals from the sea lion population will be hunted and not able to reproduce and pass on their "slow" genes. This makes the sea lion population as a whole, fitter and faster, as the faster swimming sea lions will continue to survive and be able to reproduce, further passing on their fast genes to their offspring. Therefore, sharks are naturally selecting healthier individuals to go on.
If sharks go extinct in San Diego, we will most likely have a population boom of sea lions. However, these sea lions might go after one species of fish and hunt it to extinction, due to their population size increasing. If that fish was part of some other animal's diet, than now that animal might have to look for another food source, further depleting another species, or that animal might starve to death and go extinct as well. In addition, without sharks, prey species are left unchecked and disease can spread within populations, making unhealthy individuals.
So it's not that our oceans will no longer be able to sustain life, but will rather have an unhealthy ocean productivity and less species diversity. Keep our oceans clean and speak up against shark culls. There is no scientific proof that killing sharks reduces shark attacks.
Until next time wildlife lovers!