It isn't the same because this lesson phrase is saying the subject (the colors) are not something (natural), where you are saying the subject is something (unnatural).
So just like English, the phrase would be different, although I am not completely not sure how to say unnatural in this context. Were I to guess from the dictionary definition, I think it would be los colores son innaturales.
It did accept that exact phrase as correct (the colors are unnatural).
Does Spanish actually have antonyms like this (fair/unfair, similar/dissimilar etc) or does it just use no instead? (No es fair, no es similar etc)
(Sorry, don't know the Spanish translation for fair or similar :p )
Duolingo may accept it, but you are not saying the same thing. The meaning is very nearly the same, but the translation has changed the intent of the source. In conversational Spanish I would agree that this difference generally only matters to fussy purists, but I do think you should know what you are doing because these inversions are not exactly parallel to the original meaning. This is why the antonyms exist, to shade meaning.
To answer your question, yes Spanish does have many of the same negation antonym pairings as English, and they are used with roughly the same grammar and logic.
cubrir/descubrir = "cover/uncover" and fiel/infiel = "faithful/unfaithful".
The intent of the phrase sometimes changes with the use of the antonym so you have to be careful. For an English language example, consider "faithful". To observe that someone is not faithful is very different from the accusation that they are unfaithful.
If you want to explore for other negation antonyms, I suggest this tool http://www.elmundo.es/diccionarios/ and click the de antónimos radio button.
Entonces (y yo utilizo sus ejemplos): "No es justo" es más o menos igual de "Es injusto" y "No es similar" es más o menos igual de "Es disimilar".
I'm fine with this when it makes the sentence more natural in the translated language, but otherwise, I wish it wouldn't accept differently structured sentences just because they have the same general meaning.
"Big" and "gigantic" have the same general meaning, but that doesn't mean that "big" is a correct translation for "grande".
And "The Americans are thin." doesn't mean the same thing as "They are thin Americans." :)