This is a situation that will affect too many men worldwide as they are more likely to live single / be childless. The solution is to have communities or networks of people built explitictly around the idea of caring for others.
It turns out that many animals, particularly mammals and including us, have evolved this behavior where they live with their blood relatives and take care of them, it has worked pretty well for millions of years.
Now it so happens that in many modern human societies there's a declining birth rate and consequently fewer families are being formed. South Korea's one of the worst off in this regard. You end up with a lot of people being alone and dying alone (the majority are male but women are far from immune).
Why are people not starting families? Well I haven't seen Korea-specific information but I think it's the same as most countries, people typically cite medical and financial reasons.
So basically you have to address cost of living, cost of raising a kid and rising infertility to address this, among other things.
However for a number of societies including Korea's it's probably too late. Koreans can't have more families and your solution probably won't work either. The reason is that the demographic collapse is so extreme, the population is increasingly too old to produce a new generation or take care of other people. There just aren't enough young people to go around. These societies either have to accept mass immigration or collapse. The next few decades will be interesting.