The earliest example I've been able to find (so far) in which "football fields" are used as units of linear measure is from "Jack Coffey in Flying Picture" in the [Greencastle, Indiana] Daily Banner (May 12, 1944):
Flying plane No. 4 in the picture is Jack Coffey of Greencastle, and he is shown in a close-up in a morning worship aboard the huge carrier which is as long as three football fields.
It isn't clear whether the writer is measuring from goal line to goal line or from back of the end zone to back of the end zone, but I suspect that the former is the intended sense—here and in subsequent instances of linear measurement by football field.
Three somewhat earlier instances use "football fields" as units of measure for area. From "Lost and Found!" in the [Haverstraw, New York] Rockland County Times (February 17, 1934):
Down here you'll see a tiny crack you could hardly put a knife blade into. A few minutes later it is three feet wide and you're sprinting away from it as fast as you can go. In half an hour a piece of territory as big as ten football fields, with a horrible grinding sound, has slid into the ocean, which is 1600 feet deep around these parts and darn cold. Nice stuff to sleep on. eh?
From "Kaiser Promises New Shipbuilding Record," in the San Bernardino [California] Sun (August 11, 1942):
The new plant, as large as four football fields, operates something like sub-assembly plants in the automobile and aviation industry, and feeds completed sections of ships to assemblymen on the ways.
And from "Aerial View of Orange Senior High School and Deeded Property," in the Orange [Texas] Leader (December 24, 1942):
Upper right is West End Park comprising a sodded area equivalent to three football fields complete with grandstand, bleachers, clubhouse, floodlights, etc.
Earlier still are instances in which areas are defined by how many football fields it could contain. For example, from an advertisement for Atwater Kent Radio in the DeKalb [Illinois] Daily Chronicle (November 12, 1925):
Larger than ten football fields
Here is the factory that stands behind our recommendation of Atwater Kent Sets and Radio Speakers. It covers more ground than ten football fields or sixty-four baseball diamonds.
From "Gulfport Pier," in the [Monterey, Virginia] Highland Recorder (December 22, 1939):
A pier at Gulfport, Miss., is large enough to contain six football fields.
Other countries seem have used area measurements based on the size of their football fields, as well. For example, from "Largest British Liner: The Ship Beautiful: Luxury on the Aquitania," in the Maitland [New South Wales] Weekly Mercury (May 29, 1914), reprinted from the London Daily Telegraph:
The Aquitania is just over 300 yards long, so that a walk round the ship from bow to stern and back would represent a distance of about one-third of a mile.
Her deck space is computed to be equal to that of eight football fields rolled into one.