Citizenship is not the equivalent of an American green card. In Japan, that would be Permanent Residency (永住権).

A person with no Japanese ancestry becoming a citizen is exceedingly rare. It does happen, but not very often, and the process is onerous.

Applying for PR is more common, but still a burdensome procedure and is ostensibly only granted to those who either have a direct connection to Japanese family, either through ancestry or marriage (especially marriage resulting in procreation), or who have achieved some kind of major conventionally recognized accomplishment (e.g., winning the Nobel Prize):

http://www.moj.go.jp/content/000096661.pdf

The timeframe for getting citizenship in Japan is quite long, and from what I've heard from all those I know who have it, largely not worth it - if you don't look Japanese, you will generally continue to be treated as not Japanese. Landlords who refuse to rent to foreigners on the basis that you might book it out of the country will still not rent to you, and customs/immigration workers will still consider you a foreigner. It also just potentially introduces you to a new set of bureaucratic problems, all with forms that simply don't account for the possibility of a non-Japanese name.

Japan considers citizenship to be via decent. It doesn't matter where you are born, unless your parents are Japanese you are not either. Naturalisation is an option - about 10,000 Koreans in Japan become citizens every year; out of 600,000. Obviously, it's not an easy process.

If a child would otherwise be stateless, then they will acquire Japanese citizenship if they are born in Japan. But since Korea has a similar system, Korean children aren't stateless, they are Koreans.