1. The fact that he died suddenly is unbelievable.
2. I like the idea that people can live on Mars in the future.
3. I had the feeling that he was very nice.
4. It is difficult to get the chance that you can be employed at the company.
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Are they all grammatical? Is the that clause a noun clause or an adjective clause?
Sentences 1 - 3 are fine.
In 1, 2, and 3, the "that" clauses are noun clauses, serving as appositives to the noun just before them in the sentence (fact, idea, feeling).
http://grammar.ccc.commnet.edu/grammar/clauses.htm#noun (Scroll down this webpage about 3/4ths of the way to find the explanation and examples for noun clauses.)
Sentence 4 is quite awkward. It would be better phrased like this:
It is difficult to get the chance to be employed at the company.