Is it scientifically possible? From my understandings as a biology major (for one year), absolutely. On one end, sexual maturity is jump-started by life stresses...it's an evolutionary kick-in. If this girl was under incredible duress as a child, she may have developed much earlier than most kids. The body wants to make sure it can produce offspring before it dies.
On the other end, there is
precocious puberty, which I would imagine was not treatable in 1939.
As to her having relations with someone, take a 4 year old girl and a 6-9 year old boy that might also be sexually mature...kids get curious. Maggie's links note the more sinister possibilities.