In my father's generation, people could relax after getting into university. They could party around, barely pass and still get decent jobs. Nowadays, that's definitely not the case. Getting a good job is insanely hard. You need to take care of your GPA. English certs like TOFLE/TOEIC are pretty much mandatory and you'd probably want other cert/ extra curricular experience.
Okay. So you've went through all that and you've got yourself a good job. Now that's all done, can you finally start to relax and enjoy life? Haha, no. 52-hour weekly working limit has only recently been introduced, and the current president wants to get rid of that. Even if you work your ass off, it is _very_ unlikely that you can ever afford an apartment in Seoul.
In the end, you have tired and jaded people without a fun childhood, who can't even buy their own house. I certainly would not choose to have children in these circumstances.
I've "escaped" and live a decent life outside of Korea now. But I have many friends in Korea who have "given up" (in various ways) in light of all this . There are some friends who are married and have kids, but they all have rich parents or studied/hustled really hard. In spite of all this, I miss Korea so much. But I don't think I'll be going back.
This is only one reason in declining fertility rate. There are many more (some mentioned in other comments) but this comment is already getting too ranty.
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EDIT: This comment got much more attention than I thought it would and I'd like to add some positive stuff too. There are certainly people who enjoy life in Korea by refusing to participate in the rat race and by prioitising building meaningful communities in their lives. Some things are getting better, albeit slowly, with each generation.