You could just do travel insurance, which will cover you for a lot of the big things as a lot of the small day to day stuff is cheap (even if you don't receive any national health insurance benefit).
Personally, that's what i did for the first three weeks or so until I got the national health plan (i believe it technically doesn't kick in until your registered with an ARC).
These days, as i wanted a more permanent private health option, i joined Cigna. No particular reason except i liked their Expat Health Insurance and the fact they were a large brand i knew, as well as having a Korean branch (although i think they are completely separate). I can't really talk about using it, as I haven't yet. But with a lot of the plans for most of these companies, they want a long-term commitment of a year or so I think (although i could be wrong). Overall, its about 100 000 won a month