Van Allen Belts are in part significant for their consideration in space travel conspiracies. Some argue that travel to the moon is impossible and that the United States government faked NASA's 1969 Apollo 11 mission. One argument mentioned by at least a handful of conspirators is that the Van Allen Belts surrounding the earth emit too much radiation for safe passage through them.
The Van Allen belts are the rings of charged particles surrounding the earth from one to six earth radii into space. The charged particles (protons and electrons) are held in by the earth's magnetic field. Dr. James Van Allen was the first to discover these zones of ionized particles in the earth's upper atmosphere.
The amount of radiation encountered by astronauts of moon-bound spacecraft depends on the energy and flux (the frequency of particles per unit of space) of the charged particles encountered. Most of the particles are of such low energy that they can't even penetrate an astronaut's skin, much less a spacecraft. Electrons with energies of up to one million electron volts (MeV) and protons of just over 10 MeV are also thought to be non-threatening to humans, given a low exposure time.
As for flux, about a million particles per square centimeter per second of electrons with energies over 1 MeV are encountered from 1-6 earth radii (about 6,300 - 38,000 km), and above one hundred thousand particles per square centimeter per second of protons over 10 MeV are encountered from about 1.5-2.5 earth radii (9,500 - 16,000 km). The dose of radiation for such fluxes of particles in the roughly estimated 1.5 hours it would take for a spacecraft to escape the Van Allen Belts is about 2 rem (radiation equivalent man). This low level of radiation, while a potential cause of cancer, would most certainly not be immediately threatening or instantly lethal.
So, scientifically speaking in relation to the Van Allen Belts, the Apollo missions were definitely within the realm of possibilities.