He announced "Man on the moon by the end of this decade," in about 1962 and they succeeded.
A couple of points-- first, the moon is not zero-g but rather low-g, which has a lot more advantages and fewer drawbacks for working in that zero-g. Things stay where you put them for one thing, and you have something to brace against.
It would be a good place to work out long duration tests, build an observatory, and do a few other things. As to distance, it's not as huge a deal as some might think. When it comes to supplying any outpost, the two most important factors are travel time and shipping costs. Distance is irrelevant except as it relates to those two factors.
Based on 1960's technology the moon is about 3 days away, so it wouldn't be too unreasonable to expect that early 21st century technology would at least be able to match that level of performance. A permanent moonbase would require the ability to make regular trips there, certainly at least as frequently as every 3 to six months. A reasonable safety precaution would simply be to keep moonship production/refurbishment one unit ahead of your needs. That way you would always have one craft ready to make an emergency run to the moon if need be.
It would be a great staging point for a Mars mission, and the moon would be the best possible place for an observatory.
A simple Mars trip would just set the stage for another post-Apollo retrenchment, where a moon base would be a useful step towards a permanent space presence, and be able to start bringing in a return very quickly. It would be a great space port from which to launch planetary missions-- it's been described as halfway to anywhere.
If you look at the energy costs, and use gravity whip maneuvers around Earth, you can get anywhere in the system a lot faster and more easily from the moon than direct from Earth. It's a good logical place to go next-- provided we stay.