Does 没有办法 have the same idiomatic meaning in Chinese that "No way" does in English?
In short, no. Except a selected few idioms such as 'killing two birds with one stone', where it is directly translated from English to Chinese (一石二鳥) and means exactly the same, literal translations usually do not carry any idiomatic meaning with them.

Shaw provided accurate translations to the three listed definitions of 'no way'. To supplement, 沒(有)辦法 (lit. 'no means/method') as a stand-alone phrase is used when you can't help with or do anything about something. 沒(辦)法 as a preceding adverbial phrase, however, is synonymous to 不能 (cannot; incapable of), as in you are unable to perform a certain act.



我把功課弄丟了。沒辦法之下，只好重新寫一遍。I lost my homework. There was nothing I could do but to rewrite it once more.
鄰居昨晚開派對，我沒法睡。My neighbours threw a party last night; I was unable to sleep.