One very common call was the blood relay — Greenwich Hospital might have had an urgent need for Type B+ blood, but the nearest supply was perhaps at Yale-New Haven Hospital. The State Police would be called, a trooper would pick up the blood in a cooler, drive like hell down I-95, and hand deliver the blood to waiting OR personnel. On a good day, a sufficiently motivated and skilled trooper could cover that 45-mile stretch in about half an hour. On a bad day, the trooper might end up in an accident and in need of blood himself.
That was something like 40 years ago now, and the blood distribution network has surely benefited greatly from the computer revolution, no doubt resulting in much-improved prediction of where and when blood will be needed. As a result, blood relays are probably not as common as they once were.