"Hikkikomori" is a culturally-specific psychological syndrome, affecting only people of Japanese cultural upbringing. People with this condition are withdrawn and fearful of interacting with other people, and tend to live in their parents basement well into old age. They tend not to work, shrink from education, and spend all of their time on hobbies, much to the great irritation of everyone around them. The closest way to describe them in western psychology terms would be a mix of Agoraphobia (they do not wish to leave their homes at times), avoidant personality disorder (they really don't want to interact with anyone besides their families), Autism (they tend to have very narrow, obsessive hobbies, and again the not fitting in with society thing), and extreme shyness (they find even talking to a shop clerk to buy something unbearable). Japanese psychologists claim there are up to 1 million such people in Japan. Their parents all wish they'd just move out and get a goddamn job already.
I'm imagining a city, built beneath a mountain, and having space for up to 1 million people. A train, subway style, connects this city to the rest of Japan. The city is made of little rooms cut from the stone, and has electricity, water, and Internet. There are many gloomy apartments, perfect for Hikikomori hobbies. And living here has...conditions.
For one, people living here will be charged rent. You can earn it with psychological studies on re-socializing the Hikikomori, or, we'd have a number of jobs that don't require dealing with the general public. (Socializing tends to be easier for 'Hikki" people if the other person is also one. For one, there's a greater chance of empathy in the encounter.) Many jobs would revolve around things the person could do in a small room by them-self, like programming, art, industrial design, or assembly of small objects (which would arrive and be sent back by pneumatic tube). The most extroverted position available would be store clerk, who would sell things to people feeling particularly brave that day. (I predict most goods would be sold by vending machine.) Most contact would be by internet and telephone, which these kinds of people tend to be more comfortable with than face-to-face contact.
The train would regularly go back to surface Japan, so that people could visit their famlies, and hopefully, report an improved quality of life. Japan would probably want to regularly send in psychologists, both to study the disorder and to provide therapy to make people able to function outside this little city beneath the mountain.
If this existed, it would also test a theory popular with Japanese psychologists, that "Hikkis" are the way they are because they have different ideas about independence, interdependence, and the self, which subjects them to intense bullying in Japanese society, which makes them socially withdraw. If this theory is true, then "Hikki City" would thrive. If they are, like western psychologists suspect, just really obsessive agoraphobics with varying degrees of autism, then I think "Hikki city" residents would tend to not pay their rent until forced to move back to their family's home in shame.