In baseball or cricket, playing in the field (not batting); or more generally, managing or dealing with something skillfully.
From 'field' (Old English 'feld', an open area of land), combined with '-ing' (present participle). The baseball sense emerged in the 1800s. The metaphorical sense of 'handling' or 'managing' (like 'fielding questions') developed from the idea of being ready and responsive on the field.
The phrase 'fielding questions' (meaning to answer them on the spot) comes from the idea of a good fielder being alert and reactive—they don't get to prepare, they must respond instantly, just like someone answering unexpected questions!
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