It seems a box would be cheaper and easier to make than a barrel. Was the barrel just a artists dream of having everywhere and it caught on in media?
Why barrels were and are still used:
Wine and beer were the dominant traded manufactured goods during the middle ages. It would be virtually impossible to move around 60 gallons of liquid in a box. Even if you are moving something that isn't liquid, its much easier to roll a heavy barrel than it is to push, pull or lift an equivalently heavy box.
I snagged a piece from the below AMA which talks about wine and barrel storage at least. Maybe you could reach out to that user to see if they have any more specific expertise. /u/qweniden
Also, how was wine transported in the Medieval period? Any type of pottery (except for amphora or barrels) that are indicative of alcohol consumption? "Starting in the 4th century (or so) in Europe wine was pretty much exclusively stored and transported in oak barrels. The Celts developed oak barrel technology and once the romans learned from them how to make them, that was pretty much it for the amphora. Oak barrels are both lighter and more sturdy."