You can play go for a living?
Yes, you can, but...!
In Japan, Korea and China, there are professional Go leagues, which provide for a setting in which professional Go can be played. As I have no detailed knowledge on the Chinese and Korean leagues, I shall focus on the Japanese leagues.
Japanese Leagues (pl)?
In Japan,there are two independant professional leagues, the Nihon Kiin and the Kansai Kiin.
The Nihon Kiin is the principal Japanese go association. It is based in Tokyo and was formed in July 1924. The main driver was an earthquake in 1923, which caused great financial hardship on the existing go groups. Among the innovations were the organisation of the Ooteai (big tournament) promotion system, the introduction of time limits and the introduction of amateur dan diplomas.
The Kansai Kiin is one of the professional go bodies in Japan, just like the Nihon Kiin. Shortly after World War II, the Nihon Kiin set up a branch in Kansai. The main driver for this was the increasing difficulty professionals from the Osaka region experienced during the war in travelling to Tokyo for their professional activities. When a dispute over the Honinbo title arose between the Tokyo and Osaka branches, the latter declared itself independent in September 1950 and the Kansai Kiin was born. To this day these two oranizations have remained independant, with their own roster of professionals, and tournaments.
How does one become a professional?
To become a Go professional, you have to be under 18 years of age (in special cases this limit can be extended to 25), and be one of the top 3-4 players in the yearly main round (Honsen) of the pro test tournament. But not anybody can compete in that tournament. To qualify, you must be either in the top class of the Nihon Kiin Insei, or qualify in the preliminary tournament, in which lower Insei and tournament winning amateurs participate. The test takes almost two months, but after that, if you are on top, you are a professional Go player, one of the select few who can make a living of playing go. Foreigners can compete as well, provided they meet the requirements.
What does a Go professional live off?
Teaching: Most professionals teach others go for money, some privately in clubs, some in their own schools and some teach Insei aspiring to become professionals themselves. This is the most common way of making a living.
Stipends: All professionals receive a stipend depending on their ranking in the Ooteai tournament and dan-level from the Kiin.
Titles: There are various titles that professional Go players can aspire to, for example in Japan there are: Kisei, Honinbo, Meijin, Tengen, Gosei and O-Za. To win a title, one has to win in the associated tournament in order to challenge the current title holder.
Sponsorships: Professional players may have personal sponsors, or participate in sponsored matches. Mostly, sponsors are the big newspapers.
If you want to know more about Go professionals, or Go in general, you might want to read the popular manga Hikaru no Go. This series is full of references, tips and interesting factiods on all things Go.