«Да» means «yes» only if it is the only word in the sentence/subsentence. In all other cases it means something else.
Да. — «Yes.»
Да, но... — «Yes, but...»
Он сказал «да». — «He sayed ‹yes›».
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«Да нет» is always «no», but it is not so strong as pure «нет». «Да» when it is at the beginning of the sentence is just some preposition with the meaning of «I think...»/«IMHO,...»/«I don't like it, but...»/«I'm afraid that...», and «Да»+verb means «I will»/«I want to be». Look at these phrases:
- «Да нет.» — depending on the intonation it can be «I think no.» or, if the intonation is upset, «I'm afraid no.»/«No, and I don't like that.»[/*:m:37kh1zrd]
- «Да нет, наверное...» — it's a funny phrase which can be translated as «I think no... maybe...»[/*:m:37kh1zrd]
- «Да не может быть!» — «It cannot be true from my point of view [but you just convinced me that it is true]». Compare to «Не может быть!» («It cannot be true!»).[/*:m:37kh1zrd]
- «Да что ты говоришь!» (short form is «Да ты что!») — «What are you talking about?! It is a nonsense!». Compare to «Что ты говоришь?» («I do not hear what you say.») and «О чём ты говоришь?» («What are you talking about?»).[/*:m:37kh1zrd]
- «Да что тут происходит?» — «I don't exactly know what is happening here (and I want to know!), but I don't like what is going on.». Example: mother heard that something heavy fallen at the children room but she decided to not go there. In the next moment she heard sound of broken glass. She losing patience, entering the room and saying this phrase.[/*:m:37kh1zrd]
It should be mentioned that speaker always describe his/her point of view. That is why «Да» which means «IMHO» can be omited without losing much sence. But «Да»+some_objective_fact mean upset (not «IMHO»). That is why «Да»+some_objective_fact should be translated as «I'm afraid, some_objective_fact».
«Да»+verb (do not confuse with objective fact if the verb is in past tense!):
- «Да будет так.» — «Amen.», «Let it be.»[/*:m:37kh1zrd]
- «Да пошло всё к чертям!» — «I want everything to go to the dogs / to rack and ruin / to hell.»[/*:m:37kh1zrd]
- «Да гори оно огнём!» — «I don't care if everything goes to ruin!». Compare to «Да сгорело оно всё.» («I am very upset because everything is burnt.»)[/*:m:37kh1zrd]
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So, now you know that «Да нет» is not «Yes no». And it is not funny.