Does anyone know a quiet place where I can study for very cheap or free around Seoul?

Public libraries are your best bet for free and quiet study spaces. Seoul has an excellent library system and you can find one in almost every district. Some popular ones with good facilities include Seoul Metropolitan Library near City Hall, Namsan Library, and the National Library of Korea in Seocho. You do not need a membership just to sit and study, and most have free Wi-Fi and power outlets.

Another great option is the 독서실 (reading rooms) inside community centers called 주민센터. These are usually free or very cheap and open late. Some districts also have free co-working spaces for young people called 청년공간. If you prefer a cafe atmosphere without paying much, look for places like Twosome Place or Mega Coffee during off-peak hours where a single 2,000 to 3,000 won drink can get you a seat for hours. University libraries are another option if you know someone with a student ID who can sign you in, though some like Yonsei or Korea University have open reading areas accessible to the public.