In my illustrated science fiction novel Baja Clavius: Moon Men Deep Inside you can read about men living deep inside the moon and then using time-travel devices to work on missions on Earth. I selected the moon specifically as the location of an underground base where the fictional time travel agents live. The reason was that time travel as I imaged it will not work on Earth but works perfectly in the low lunar gravity.
However, men (or women) living on the moon is presently stuck squarely within the realm of science fiction and remains a great distance away from becoming science fact. Establishing a permanent human colony on the moon has often been imagined to be a practical outcome of the round-trip lunar missions men made from the United States starting in 1969 and ending in 1972. My storytelling about men living deep inside the moon is set in the 23rd century in a future civilization which I envision has many technological advancements that are unknown in the present day. But it is relatively easy to write science fiction stories.
Developing advanced, real-world technologies is a whole other matter. Even if the United States (or any other world power) wanted and could pay for a permanent lunar colony today, there just is no present-day technology to make practical and safe human travel from Earth to the moon and back again. History showed how the United States successfully maintained rockets and spacecraft in the late 20th century which made possible voyages to and from the moon. This turned out to be a very short-lived manned lunar exploration program, however. Those specific technologies (such as the massive Saturn rockets and the Apollo spacecraft) were extraordinarily costly and it should come as no surprise that those technologies no longer exist today largely because of how expensive they were. And nobody on Earth has yet to develop a cost-effective, practical, and safe way to get people to and from the moon on an ongoing basis.
Solving the human transportation challenges between Earth and the moon must come first before a permanent human colony could be built on the moon. But next there will come communications challenges if people on the moon intend to stay connected in real time with people on Earth. Due to the speed limit for how fast light travels taken together with the physical distance of the moon from Earth, a communications signal needs 1.28 seconds to travel one-way. This mean seem merely like a “slight” delay, but if the need is to have remote-controlled devices on the moon which are operated by people on Earth, that delay introduces many potentially deadly risks. Any remote-controlled operations on the moon that could have life or death consequences for humans living there would need to be avoided due to that “slight” delay. The most practical and cost-effective solution would be to set up the human colony on the moon as fully autonomous which at a minimum would rule out the use of essential, life-or-death signals (such as to operate remote-controlled devices) to travel round-trip from Earth to the moon. Who has the authority to govern that autonomous colony to maintain ethics, morality, privacy, law, and judicial resolutions of conflict?
The other serious challenge to humans living on the moon full-time is the existence of radiation on the lunar surface that can injure and kill humans. The most apparent solution would be to build human housing structures under the lunar surface located at a sufficient depth that could shield humans from the effects of radiation. The concept of an underground human colony on the moon is what I used in Baja Clavius: Moon Men Deep Inside, but I did not deal specifically with the financial costs of building the time travel base under Clavius crater. No doubt the cost of creating and maintaining underground human housing and work areas would be hugely significant even assuming modular structures could be built on the lunar surface and then lowered down under the surface after excavations had been completed. This high cost factor means that the expected outcomes of having a permanent human colony on the moon would need to be mission-critical for the entire human civilization on Earth. The expense of the Apollo moon program was extraordinary, but President John F. Kennedy had in 1961 urged the federal government to land men on the moon and return them safely to Earth before the 1960s were over. Kennedy was assassinated in 1963. The American manned moon exploration program largely was deemed mission-critical stemming from the murder of JFK.
A fictional mission-critical situation I created in Baja Clavius: Moon Men Deep Inside was the imperative to send time-travel agents to the past to change timelines for the specific purpose of preventing what would otherwise be the inevitable self-destruction of civilization on Earth.
Two related obstacles hindering humans from living permanently on the moon are water and waste management. Someone will need to discover ways to use what is already on the lunar surface (or underground) to harvest drinkable water for human colonists. While there have been recent, encouraging developments following the discovery of water on the moon, the quantities needed for a human colony probably are not going to be met by using the available water on the moon. This means new, more sophisticated technology will be needed to replicate our most essential fluid on the moon and produce synthetic water. New, more sophisticated technology to recycle human waste (liquid and solid) on the moon that can be used in part to produce water safe for human consumption will also be an essential requirement.
Keeping humans fed while they live in a lunar colony is another major challenge. All consumable food substances will need to be grown and/or processed underground since there growing food substances on the lunar surface is unrealistic and impractical. Food supplies cannot be shipped to the moon from Earth due to the high costs of doing so.
All of these known challenges when combined with any challenges we have yet to discover suggest that humans living on (or below the surface of) the moon certainly will be one of the most significant social and technological advancement of our human species if that day ever comes.