Hmm, it's just past tense. 있었다 = I had/there existed (something), that I/there may or may not still have/exist. Same with 없었다. 하고 있었다 = was doing something, with emphasis on the time span it occupied more than the action itself (I think). For example 전화가 왔을 때 티비를 보고 있었다. "When I got (lit. there came) a phone call, I was watching TV."

It’s a hard feeling to explain. I understand it in already written text, but when I’m having a conversation in Korean, the fact that it’s in past tense throws me off.

있다 has a few different common meanings. 1) there is/exists. 펜이 있다. 펜이 없다. 나 오늘 중요한 일이 있어. 2) to stay at a place. 나는 여기 있을게 = I will stay here. 가만히 있어 봐 = Be still for a second. 3) to pass/wait (an amount of) time. 일주일 더 있으면 방학이다 = One more week and it's school break. And 없다 has the meaning of #1 negated. So -하고 있다 means "is doing something" and -하고 있었다 "was doing something" as you said. e.g. - 그림을 그리고 있었어 = I was/have been painting. - 나는 티브이를 보고 있었어 = I was watching TV. - 지금 뭐하고 있어? = What are you doing now? - 시간이 없다 = There is no time. - 어떻게도 할 수가 없었어 = I couldn't do anything / There was nothing I could do. - 외딴 역의 대합실엔 아무도 없었다 = There was no one in the waiting room of a remote station.

You understand the meaning, but it throws you off? The sounds a little strange to me.