Neil Armstrong's first step on the moon proved that reaching the moon safely was possible. All that is necessary is the political, economic, and public will to support the endeavor.
A new frontier is achievable
Armstrong's mission proved that existing science could successfully land a man on the moon and return him home safely again, even when using the technology of the 1960s. Modern technology should be capable of at least as much.
It also proved that mankind was capable of surviving far outside the Earth's gravity and radiation-protected zone. No one knew that for sure until someone tried it. There had been no previous human experience on which to base a guess. Armstrong's safe return and subsequent long life proved that mankind could fly into space and set foot on unprotected new worlds without succumbing immediately to space-related health issues.
However, some space-related health problems have been identified after that historic mission. So far, there are 3 known space-related health problems: space-related cataracts, bone loss, and a lifetime increased risk for cancer. Bone loss is related to loss of gravity, while the others are related to increased radiation exposure when going outside the protection of the Van Allen Belt.
All of these problems are serious, but bone loss and space-related cataracts may directly affect future astronauts during a years-long mission in deep space, such as the proposed mission to Mars. These health issues must be solved prior to any long-term space exploration.
Where there is a will, there is a way
The knowledge that this kind of feat can be done means nothing without the political, economic, and public will to follow up on that feat. Mankind has not yet taken a second step which could be considered a great leap. Some might even say that mankind has taken an enormous step backwards.
Apart from some political rhetoric, there is no real indication that a second step is imminent. In fact, many would consider that the constant budget cuts to NASA represent a step back from Armstrong's pioneering first step. Outside the space industry and related academic and technological fields, which stand to gain financially, there is very little economic will to spend money on space exploration.
On the other hand, private enterprise has started to step in where government will and money is lacking. The first space plane broke the barrier of space in 2011. Virgin Galactic is already booking passengers for "space cruises." Maybe the future no longer belongs to any government, but to private citizens who are willing to invest the money to make it possible.