An '03A3 in that condition was almost certainly a DCM gun. They sold for $14.50, but are probably worth a bit more now
Having the headspace checked is a good idea, but it is extremely unlikely that there is any problem. Production of the M1903 and M1903A3 was cranked up in WWII when it looked like production of the M1 rifle would not be able to meet the need. By the time the later M1903A3's came off the line, there were more than enough M1's to go around and the bolt rifles went from the factory direct to depots and were never issued. After the war, many were given to U.S. allies, and many more, brand new, were sold to NRA members through the Director of Civilian Marksmanship.
You see, back then, the U.S. had just fought a war against tyranny and a free nation actually trusted its people with military weapons.