This needs to be done in a single game and will likely be achieved late in the game. You will first need to ensure you have the capacity to hold 250,000 barrels and can check this by viewing the first page of your journal. To increase your total capacity you either need to build additional warehouses or purchase/capture additional ships.
An alternate strategy is to have at least 15+ businesses running. Save your game as this method will cause you to lose a large amount of money over time. Fire your steward to prevent the goods being sold to the local market, then hold down to sped time up. You can amass the 250,000 required barrels, even if you don't have the storage, as barrels will continue to amass beyond a warehouse's total capacity. The more businesses you have, the quicker this will be. Just be sure to reload your game as you will no doubt amass significant warehouse charges doing this method.
TIP: When you are going for the achievement, it is a good idea to check the taverns in towns you have established warehouses in, as there will be the occasional merchant or pirate who will offer you a large number of goods for a price and these can quickly boost your total.