Verbs with 다? (없다고, 있었다는)
Is this a correct translation for these sentences? What does ~다~ in 없다고 and 있었다는 mean?

1) 여섯 살 적에 어른들이 화가로 출세할 수 없다고 나를 낙심시켰기 때문에 나는 … 그리는 연습을 하지 않았으니까 말이다.
“It is because i haven’t done any drawing exercises, since when i was six adults said that i won’t succeed as an artist and discouraged me”

2) 내가 사람 사는 고장에서 수천 마일 떨어진 곳에 있었다는 사실을 여러분은 잊지 말아 주길 바란다.
“I want you all to not forget the fact that i was in a place thousands of miles removed from areas where people live”

Question

Verbs with 다? (없다고, 있었다는)
Is this a correct translation for these sentences? What does ~다~ in 없다고 and 있었다는 mean?

1) 여섯 살 적에 어른들이 화가로 출세할 수 없다고 나를 낙심시켰기 때문에 나는 … 그리는 연습을 하지 않았으니까 말이다.
“It is because i haven’t done any drawing exercises, since when i was six adults said that i won’t succeed as an artist and discouraged me”

2) 내가 사람 사는 고장에서 수천 마일 떨어진 곳에 있었다는 사실을 여러분은 잊지 말아 주길 바란다. 
“I want you all to not forget the fact that i was in a place thousands of miles removed from areas where people live”

Answer

It is  a correct translation for these sentences.

-다고 or -다는 is used for indirect quotation

NOUN + -라고 → VERB + -다고
NOUN + -(이)라는 + NOUN → VERB + -(ㄴ/는/었)다는 + NOUN

VERB + -(ㄴ/는/았)다고 + 말하다/하다/쓰다/적다/부르다/etc

Some examples)
뭐라고 했어요?
= What did you say?
전화 온다고 했어요.
= I said that your phone is ringing.
그 사람이 내일 온다고 해요.
= He says that he will come tomorrow.
그 사람이 언제 온다고 했어요?
= When did he say he would come?
그 사람이 이거 뭐라고 했어요?
= What did he say this was?
한국은 겨울에 정말 춥다고 들었어요.
= I heard that it’s very cold in winter in Korea.
저도 간다는 메시지를 남겼어요.
= I left a message saying that that I would go too.
한국어를 공부한다는 내용이에요.
= It’s a story/content that (someone) is studying Korean.
경은 씨가 제주도에 간다는 이야기를 들었어요.
= I heard (a story that) someone (was) saying that Gyeong-eun was going to Jeju Island.
지금 간다는 사람이 없어요.
= There is no one that says that they will go

화가로 출세할 수 없다고 어른들이 낙심시켰다
Adults discouraged me by saying that I won’t succeed as an artist.

수천 마일 떨어진 곳에 있었다는 사실
the fact that i was in a place thousands of miles apart

Also refer to 
https://www.italki.com/question/318096
https://www.italki.com/question/307738