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In 5e RAW, a character with 20+ STR could carry one 300-lb barrel so long as they have nothing else worn or in their inventory. They'd be naked. (Carrying capacity is 15 x STR.)

A character with 10+ STR could drag a 300-lb barrel around, but not lift it. (Pushing/dragging capacity is 30 x STR.) I would argue that this would make it (nearly) impossible to position that barrel stealthily.