A quarter is 808.5 mm3. If you melted the quarters into a slurry, and poured that into the barrel, you could fit 208 million/808.5 = 257,000 quarters into a 55 gallon drum.
If you want to preserve them whole, this becomes a stacking problem. There will be spaces between the quarters, and we would have to estimate how much space is wasted. Just drawing packed circles on a piece of paper I'd guess perhaps 10% of the volume would be air, so deducting 25,000 quarters from our solid mass gives a ballpark estimate of 225,000 quarters.
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Since 1000 US gallons = 4000 US quart, then answer is 4000 quarts.
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That sounds like an excellent math experiment - see how many quarters it takes to fill one inch of the can - then multiply out how many inches tall the can is so you can know about how many quarters it will take to fill the whole can!
i just emptied one the other day and bought it to the bank...$3748 total