Can a foreigner open up a business in Korea?

It is very much possible to open a business as a foreigner, however you will have to have the right visa. Those who possess residential visas like F-2, F-4, F-5, F-6 visas, will be able to start a business with no limitations, just like any Korean national.
 
If you have a visa for foreign entrepreneurs (D visas) you can also start a company. In order to be incorporating a limited liability company – you need at least 100M KRW (about 90,000 USD) and at least one director and one shareholder(who don’t have to be Korean). 
 
This is a very good detailed read on how to get the right visas and licenses for entrepreneurship in Korea. Highly suggest you look through it and hope this helped!
https://shieldgeo.com/setting-up-a-company-in-south-korea/