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.
Ok so this gets into something specific about barrels, where you wouldn't drag it, you would roll it. Which, theoretically you could do that stealthily - under the right conditions.
However, as DM I would insist - and this is SUPER important - that you play the right Theme Music while doing it.
Otherwise you go at disadvantage. :P
While you can roll a barrel around, it's still going to be unwieldy and loud. You certainly aren't going to be ducking between shadows with this thing. Are there situations where you could move unnoticed with one? Probably, but I imagine they'd be uncommon.
Also, the appropriate theme music for BG3 is this.
