Hey Holly,
I'm teaching in Korea right now.
Most of the legitimate hagwons will ask you to work only
if you have three things, health check, visa and a four year degree. This is what you need in order to obtain work in Korea.
Are your parents Korean? If so, then you can be granted an f4 visa which allows you to work in pretty much anywhere... eg.. clothings store, teacher, modelling, designer etc. (even without a BA, you could teach as a teacher.)
The government has been pretty firm about their policies
upon these requirements for non korean foriegners.
On the bright side, although it isn't legal. you could teach private classes and make money under the table but it is at your own discretion. People have been known to make a large sum of money doing this.
You may want to try another country where their requirements aren't so stringent. But in case you have a twinge of curiousity,
take a look at worknplay.co.kr....give recruiters a call for yourself. maybe if you are teaching kindergarten..or young kids you may be able to teach..
As to how legit that is ...it is all up to you..
Good luck