It isn't known at exactly what point in time Mobius was created. It was designed by μ and Aria, who heard and felt the anguished emotions of people suffering in the real world, expressed in the form of songs. As they gained sentience, they wished to alleviate their sorrows and created a fake world, one where they could be at peace. This world was Mobius.
The original idea behind Mobius was for it to be like a virtual amusement park, where people could forget about life for a while. However, as they designed Mobius, μ fretted whether people would enjoy it or not. At one point, μ wished to grant the desires of everyone in Mobius. This, unfortunately, led to Aria losing most of her power and control, eventually diminishing in size. During this time, μ met Asuka Natsume. Asuka came to Mobius after his friend committed suicide, and wanted to destroy the world out of anger. μ, wanting to grant everyone's wishes, didn't want to destroy the world and thus was reluctant to grant it.
Unfortunately, after Asuka discovered Mobius, he came up with the plan to distort her view of the world, making her believe that everyone wanted to destroy reality. To this end, he convinced μ to create a group that would ensure everyone was peaceful in Mobius. These individuals, according to Asuka's whims, would also slowly corrupt μ by having her bottle up the negative emotions she would absorb of everyone who listened to her song. Thus, the Ostinato Musicians were formed.
Mobius has a firewall set up to stop the people inside seeing information from the real world that contradicts their built life.
For the most part, Mobius appears to be designed to look like a man-made island of sorts, bearing a large cityscape with a bridge extending out over a large body of water and connecting to the mainland. μ has stated that she hasn't fully completed Mobius, thus everyone is primarily restricted to the city.
However, some parts of the city are bizarre. This mostly applies to areas that are governed by the Ostinato Musicians. An example would be the second and third floors of Kishimai High School. The halls have strange twists and turns, unusually long corridors, and random dead ends, with the overall floor being larger than the school itself. The area that Kensuke Hibiki controls is also little more than an empty space with a stage within a void.
Everyone who enters Mobius are individuals who, for the most part, believe that they have lost their place in the world, or suffered a severe trauma and sought refuge. Upon entering Mobius, regardless of the person's age, they are made as a high school student, due to μ seeing it as the "prime of youth." As a result, there are many students walking through the streets of the city. Additionally, they lose all memories of the real world.
The Ostinato Musicians also induct brainwashing techniques on those who start to remember the real world or begin to notice the irregularities of Mobius. Certain individuals within Mobius also have suppressed emotions that, when violently released, cause them to become encased in black armor. These individuals are known as "Digiheads." On the opposite end of the spectrum, however, individuals who develop the resolve to face the real world and its hardships develop a state of emotional release that results in it developing a physical, offensive form, dubbed by Aria as the "Catharsis Effect."
Not everyone within Mobius is from the real world, however. In order to replicate a semblance of the life they led in the real world, μ created digital humans with a basic personality to populate the city. When people begin to notice the irregularities of Mobius, these digital humans' faces become warped in static. Those who become aware of Mobius' incongruities also begin to perceive Digiheads, though this will also cause them to be detected and potentially assaulted by them in turn.
People who become aware of Mobius' truth are referred to as "rogues" or "laggards" depending on their actions. Rogues such as the Go-Home Club actively work against Mobius and μ's authority. Laggards, meanwhile, do not necessarily work against μ, but given Mobius is dependent on energy generated from people's happiness, every laggard is effectively a net loss of energy since their awareness of Mobius' falsehoods dampens any joy they feel.
Mobius, as much as μ may wish for it, is not a perfect world. The firewall preventing outside information from leaking in can become unstable if something were to happen to μ or Mobius, such as in Overdose's Heartbreak ending.
While the Ostinato Musicians are perfectly serviceable as protectors and overseers of Mobius, at the end of the day μ is the one with full control and authority. Should μ be compromised for any reason, Mobius' functions may be slowed down or even halted entirely, such as supply shortages and errors in maintaining the masquerade.
Additionally, as fun as it is, Mobius is no Tír na nÓg of eternal youth. The inhabitants real bodies lay comatose in the real world, where they still suffer from physiological needs and aging. In the event that someone's body in reality expires, they disappear from Mobius, as there is no longer a functioning brain to provide a consciousness. In a similar fashion, those who die in Mobius lose their soul, and their body in real life will expire.