N2
ようがない・ようもない
There's no way to, It's impossible to
車も自転車も壊れているので買い物に行きようがない。
Because my car and bicycle are broken, there is no way for me to go shopping.
Explanation
「ようがない」 means 'even if I wanted to, there is no way to', because the method, clue or condition is missing. It attaches to the verb stem: 「連絡先を知らないから、連絡のしようがない」 (I don't know the contact details, so I can't reach them), 「ここまで壊れたら、直しようがない」 (Once it's this broken, it can't be fixed). With nouns: 「noun+のしようがない」. 「ようもない」 is stronger.
Mixed up with
「ようがない」: no means or conditions, so you can't do it even if you want to; a feeling of helplessness.
「できない」: simply cannot; could be lack of ability or a rule against it.
「わけがない」: impossible by common sense.