Answer: A) The distant past.
Step-by-step explanation: Based on the information provided in the given sentence, the best definition of the expression "remote antiquity" is the corresponding to option A: distant past, we can know that by the context of the phrase, because the man wore a coat "dating back" to a remote antiquity (the phrase "dating back" gives the idea of a time in the past), and also by the definition of the individual words: antiquity means "from the ancient past" and remote means "distant."