The phrase implies a task is as impossible as fitting too much liquid into a small container. Expecting one person to do the work of four is like trying to put a quart into a pint pot.
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Cultural Origin
It's a British phrase, so it may not be understood in countries using different liquid measurements. He described his new job at the United Nations as trying to put a quart into a pint pot.
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Metaphorical Use
Often used metaphorically for tasks with inadequate resources or space. The new software has so many features, it feels like they've tried to put a quart into a pint pot.