Gyeongju is a nice city in South Korea full of historical sites to visit. Especially known for the Bulguksa Temple and the Seokguram Buddha Statue, tons of tombs and the Anapji Pond, and beautiful Hanok Village (Traditionnal village) in the north.
(Updated 31/12/2017) Go to the road 35 in the south of Gyeongju, which leads you to the entrance of the highway. There is a GS gas station here. You can stand at the entrance of the gas station on the grass spot which forms like a little "hill" so people who were not supposed to go to the station will see you and will be able to get in the gas station to pick you up safely. You NEED a sign here as this is the entrance to the highway to go to the South towards Busan and Ulsan AND towards the West to Daegu then Jeonju, Seoul...
You can also keep walking and probably reach the tollgate to hitch from there.
Here is the spot