As Kayla Kwon has indicated, there are several family names that are two characters.
In my comment above, I gave two examples of people that I knew that only had a single character given name. I knew a woman named 양순 (Family name 양, given name 순) as well as a man named 권혁 (Family name 권, given name 혁). While somewhat rare, names of a single character are out there.
A single character given name was actually somewhat common among historical scholars. Men like 이이 and 이황 come to mind. They also had pen names (율곡 and 퇴계) that were only two characters.
I have also seen people with THREE characters in their given name. One person I knew had the name 최진주성 (Family name of 최, given name of 진주성). Although not true in this case I have given, I believe that it is more common to see a "long" given name if the name is pure Korean (and not Sino-Korean).
As I understand it, the South Korean government limits a name to a total of five characters. (The link is in Korean. The final paragraph speaks of the limit of five characters).