1. Is there a big market around here?
1. Yes, there's one near here.
2. Yes, there's a big one near here.
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Q1: Are both answere grammatical? Which one is commonly used?
Q2: Does 'one' in #1 mean 'a market' or 'a big market'?
Both answers are fine, yes. Answer 1 is probably more commonly used.
And yes, "one" is being used to refer to "market" in both sentences.