Not every country measures oil in barrels, but that particular measurement is still popular in the United States, which means it remains in the public vernacular even if it has largely lost its significance in a mostly metric world economy. At one point in history, oil producers did store oil in barrels, although the size and nature of those barrels were far from standard.
When the first oil fields were tapped in Pennsylvania during the 1860s, there were no steel 55 gallon drums in which to store the oil. Instead, the oil was pumped into whatever containers could be found, including pickle barrels, cracker barrels and whiskey barrels. There was no standard size oil barrel, but eventually the wooden whiskey barrel became the most popular storage container to hold crude oil until it could be shipped to be refined.
The standard whiskey barrel at the time held approximately 40 gallons of liquid. Early oil producers wanted to ensure their customers received every last drop they ordered, so they actually overfilled the barrels to 42 gallons. This 42 US gallon mark (which is about 35 Imperial gallons and about 160 liters) became the standard measurement of oil in barrels produced in American oil wells.
Eventually the wooden whiskey barrels gave way to steel drums which provided more protection against leakage and contamination. Although these steel drums were designed to hold 55 US gallons of oil, the standard 42 US gallon barrel is still considered to be the correct legal measurement of oil in barrels. When oil producers or economists speak of the number of barrels of oil produced in Saudi Arabia per day, for example, they are applying an American measurement, not one the Saudis themselves might use.
The reason other oil-producing countries rarely use the term "barrels" to measure their production rates is because they rarely store their products in actual barrels anymore. The oil pumped out of the ground is more commonly transported in large tanker trucks or through elaborate pipelines directly to the refineries, or massive cargo ships for overseas delivery.
Individual companies may store oil in barrels, but the largest commercial oil producers rarely do unless the product is going to be shipped to a remote location, such as a military base or third world countries without storage facilities. It is far more likely to see a derivative of crude oil, such as gasoline or kerosene, actually stored in steel drums or barrels.
Therefore, the reason we measure oil in barrels is mostly to provide a familiar reference for those who grew up with images of actual oil-filled barrels rolling down a conveyor belt. In reality, only a percentage of a barrel of oil is converted to gasoline or petrol, so a number such as 1,000,000 barrels of crude oil does not necessarily translate to a surplus supply of gasoline. It only refers to the number of gallons of raw crude produced that day, not how much has been refined into various petroleum products.