"No way" in English has a couple of meanings.
Like, it can be used to express surprise. Does 没有办法 also carry some idiomatic meanings?
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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.
"没有办法" is actually a bit different from "no way".
The examples from your link:
No way will he beat me one-on-one. 他不可能单挑赢过我
No way! It's Janet's turn. 想都别想！该轮到Janet了
No way! That's great news! 不会吧！这真是天大的好消息！
(If those are the idiomatic expression you are looking for, but definitely not the idioms as "成语")
"没有办法" goes more like "no idea" (i.e. I tried but no idea how to do),
"不行/不可能" goes more like "impossible/no way" (i.e. I won't do that).