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There's no way I could do something like that in public.
Literal
Public [at-で] that-kind-of thing can-do [reason-はず][also-も] not-exist.
The pattern ~はずもない is an emphatic version of ~はずがない ('there's no way, cannot be the case'). The も replaces/adds to が for extra emphasis — 'not even the possibility exists.' 人前 ('in front of people, in public') literally 'before people.' そんなこと 'such a thing, that kind of thing' — here dismissive. 出来る is the potential form of する.