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There's no way something this big is going to fit on a bike's rear rack.
Literal
Like-this big [nominalizer-の] [subject-が] bike [gen-の] cargo-rack [on-に] ride [no-way-わけない] [right-だろ].
The fixed pattern ~わけない / ~わけがない expresses strong denial — 'there's no way / there's no logical basis for X.' わけ is the noun 'reason, logical basis,' so わけがない literally 'there's no reason [to think X].' Blunt masculine だろ seeks reluctant agreement. デカイ is the casual slang form of 大きい ('big'), especially in masculine speech. チャリ is the casual slang for 自転車 ('bicycle'). The whole sentence is vivid slangy speech resisting an unreasonable demand.