Well, the advantage to the gyms at the link blabl shared is that the general public, including foreigners, are allowed in and many charge a per-visit fee although perhaps not all do. The cost is often subsidised, so you can sometimes end up paying as little as 1000 won or 1500 won per visit. Again, it depends on the facility and on the municipality. All such gyms have multiple clubs operating within but you can decide for yourself whether to join and not joining a club doesn’t mean you can’t enter the gym and play. It’s just easier to get people to play with you if you join one of the clubs, the fees of which usually go to social events rather than sport. If you play at a school, your club fees will help cover the cost of renting the school gym but that’s not true at a municipal facility.
Dosol is not too far from you but it is 7km from the KAIST main building so it’s a bit of a hike. Namseon is only 6km and is likely the closest to you from the list of public gyms. If you mean bicycle, you might like this one because much of the distance can be covered via the riverside bike paths.
From the Badmintontimes list, #13 plays at Eoeun Elementary School, which is quite close to you. #1 is at the Daedeok Research Centre or whatever it’s called, just east of Expo. 3 courts and I find it hard to believe the public could access it but I’m not sure.
Actually, one of the closest might be #1 on this list. That club plays at the Galma Dong Olympic Memorial Training Centre or something. It’s about 3km from you. It says they play in the morning but you can visit the gym to see if there are other times. On the same list, #3 plays at Baekun Gymnasium, which is about 5km from KAIST. That also might be a public facility. I haven’t been there.
On the other hand, if you are actually at that other KAIST campus, Munji, then the Jinjam gym will be closest to you. Good luck.