Non-teaching jobs are not especially easy to come by in Korea, and generally require an exceptionally high level of education, fluency in Korean, a business partner, or personal wealth. Part-time work is also technically illegal, but this does not stop people from doing the odd job here and there in freelance translating, acting, and so on.
To work for a foreign company in Korea is somewhat difficult, as they are more likely to send somebody from within the company to Korea rather than pick up a foreigner living in Korea. Working for a Korean company is somewhat easier, but usually much lower paying. People in fields such as IT, certain types of electrical engineering, and technical writing can find jobs.
ProKorea.Club is a newer site that combines non-teaching jobs in English from all the major Korean Job Portals. It automatically updates with the latest job postings (that were posted in English) from Saramin, JobKorea, and PeopleNJob every hour.
JobSeekr is the highest quality job site for foreigners and Koreans with diverse linguistic and cultural backgrounds. After making a profile you can connect with other professionals and apply to jobs in one-click. The jobs on JobSeekr typically provide higher salaries and non-teaching positions not found elsewhere.
Korea Job Finder is one site that has non-teaching jobs for foreigners. You can search for "non-teaching jobs" at the top of the page to view all positions that are currently available. Worknplay also posts non-teaching jobs like editor or manager positions in Korea.
http://seoul.craigslist.co.kr/ is also a good resource if you're looking for a specific type of job.
Job Hunt Korea is a recruitment site for English teachers coming to or already in Korea. They have a phone number listed on their site to receive additional information.
Visas for doing jobs other than teaching are H-1 (Working Holiday), D-8 (Business Visa), E-7 and E-3
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