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... Here is the spot on map
You can also keep walking and probably reach the tollgate to hitch from there. Experience summer 2018 from savvyhitcher: I reached the tollgate on foot from the center of the city. It is a good 45 minutes walk depending on how fast you walk, so be prepared. I did my sign at the gas station but after 10 minutes no one picked me up so I left. I also tried the hilly spot for 5 minutes and no one stopped. It probably would have worked but my intuition told me to walk towards the tollgate. On the way there are apricot trees which make for a delicious snack. The staff let me hitch at the tollgate without any problem and seemed not to care at all to see me (a relief after the bad experience in Daegu), as if he had seen many other hitchhikers before. It took me roughly 10 minutes to get a ride from the tollgate.