I read somewhere on the internet that it is possible to do this by rail. I have pasted the info I found below.
Could anyone confirm the first option, namely a slow early morning train?
"Option 1: the direct route. Apparently there is a very early train in the morning once daily that does this route. It’s a slow train, but I dont mind that when I’m travelling, fun to see the country. I arrived at the station at 9am however, and it had already left. So check this out the night before if it interests you.
Option 2: KTX route. If you use the KTX system, the key is you have to actually change railway stations in Daejeon (from Daejeon to Seo Daejeon). That’s part of the reason that makes it so confusing, technically you can’t go KTX from Busan to Mokpo direct, since it’s not connected. But if you are happy to take a bus/taxi for a few kilometers to switch stations, then you can do it. And it’s better than going via Seoul."