A robo-car with shower, kitchenette, bedroom would radically transform the situation for poor people worldwide.
Edit for response: a robocar with all of these amenities would be an RV costing much more than $10,000; the land cost would just be shifted to the public as these vehicles take up all available parking. Not to mention that a shower and kitchenette would require utility hookups, meaning they would really need an RV camp...
Even in slums in India people are capable of affording low cost cars like tata nano's. And with the great efficiencies brought about by robo-cars, they would be even cheaper to manufacture and sell.
And the above doesn't factor in the value of land. Much of rent/mortgage goes into paying off the actual land-owner (usually a bank). Take away that exorbitant cost and robo-cars would actually be much cheaper than even a small apartment. At $700/month, $8,400 per year VS $10,000 one-time cost robo-car.
(Admittedly, there may be some cheaper subsidized rates for charging electric cars today, but once everyone has one, will they be able to keep those subsidies?)
Also, it'll be awesome trying to drive anywhere with the additional traffic density.
If one could convince legislators and voters that people should not be forbidden to live in cars, than the original ordinances against parking can be revoked.
The situation you described, 'build cheap apartments', is already the current situation. It hasn't worked has it?
Land ownership is a heavily regulated industry, and wherever you have regulations you get distortions in the market caused by political activity (electrician/plumber unions, local land owners etc).