In the 60's there was a space race. The USA wanted to put someone on the moon. It was accomplished. However not long afterwards the STS project was launched.
The STS project was to replace the saturn rockets with a re-usable launch vehicle for advanced laboratory space studies. As a re-usable engine it was pitched as being cheaper, reusable, etc. It had drawbacks, not being able to reach escape velocity from the surface of the Earth is the primary one that keeps it from being viable for a moon mission.
There was significant opposition to the STS program both within and without NASA. Once design and plans had been laid in order to make sure the STS platform succeeded the saturn rocket designs and systems were gathered up and destroyed. The theory was that as long as people had a fallback, they would emperil the STS program. There were significant monies tied in STS and certain members of the government took steps to ensure there were no alternatives.
The USSR, never made it to the moon, because they lacked the required navigational system.. in a way this was geniuses working against each other and the US simply won at navigation. The USA built the correct propulsion units (saturn rockets), and those were stable enough to be used. The USA also had the significantly better navigation system then the USSR was using.
Some have mentioned the VARB, (Van Allen Rad belt) but that was not a factor. The radiation belt itself was disregarded by Van Allen even if you went directly through it. Also NASA timed the launches and made sure to skirt the belt. It was not something unknown at the time.
Edit: Adding info to deal with original questions's followup: Radiation was a definite concern for NASA before the first space flights, but they invested a great deal of research into it and determined the hazard was minimal. It took Apollo about an hour to pass through the radiation belts - once on the outbound trip and once again on the return trip. The total radiation dose received by the astronauts was about one rem. A person will experience radiation sickness with a dose of 100-200 rem, and death with a dose of 300+ rem.
Additional information can be found here: http://www.braeunig.us/space/hoax.htm
Just for your knowledge, the radiation in your dental chair will probably also not kill you. It will make you MORE likely to develop cancer. Hospitals and Dentists know you would sue them to death if they didn't take those precautions.
To this day, there is no feasible way to reach the moon without combinations of technology. We don't have the saturn rockets anymore and while I believe at least someone may have designs.. no one knows who and where. However the russian proton rockets do exist, and are the only rocket vehicles strong enough to make the escape of Earth gravity from the Earth's surface. I believe they are a little unreliable, and I would assume that the Russian navigation system has come up to speed, although the USA may still be ahead there.
I find it extremely likely we went to the moon. The Saturn V rocket was measured and its velocity and payload (3-6T) were among the only capable of escaping Earth's gravitational pull. To imply that we would/could not field the hot shots willing to risk this would be incorrect. We did field them. To put it bluntly, we had the money, we had the astronauts, we had the rockets, we had the navigation system, and we had the payload. In short, there was very little reason to lie.
Honestly, if you believed this was a hoax. Do you really think the russians would have stopped unless they really thought it was true? They would have claimed it's an American hoax and continued progress to reach the moon.
We landed on the moon. There just is not a lot of reason to go back, unless we are going to establish something permanent, which is problematic.