The Verdict: Not eligible
Korea passes citizenship only from a parent to a child, so a Korean grandparent or earlier ancestor is not enough on its own. You can still live and work in Korea on the overseas Korean visa, and naturalise later.
Article 2 of the Nationality Act looks only at whether your father or mother was Korean when you were born. No naturalisation category is based on a grandparent.
The overseas Korean visa (F-4) is open to anyone who held Korean citizenship by birth and to their descendants in any generation, once they hold another citizenship. It lets you live in Korea for up to three years at a time, renewable, and work in most jobs. After two years on it you can apply for permanent residence (F-5-6) if you meet an income or other test, and permanent residence is the first step to ordinary naturalisation. Our Diaspora quiz for South Korea covers the visa.
If your ancestor is officially recognised as an independence patriot, their descendants in any generation can apply for special naturalisation and keep their other citizenship. Our Historical Displacement quiz for South Korea covers that route.