The woman in question is Eleanor Johnson(Ann Sheridan), husband of Frank Johnson(Ross Elliott), and while the title may imply this woman is running from something, she's actually running towards someone; her husband. Walking the dog one night, Frank witnesses a murder, and is shot at himself before the killer hops back into his car and races away. When the cops arrive, Frank learns that the victim was a witness in a federal case involving organized crime, and now he's stepped into that position, as his identification of the murderer will somehow still allow the prosecution to go forward. Not wanting to be put in the spotlight, and possibly become a murder victim himself, Frank seizes an opportunity to run for it and go into hiding.
To be honest I had a few problems with this opening. The look and staging of it was all great, but the logic felt a little forced, the characters a tad too dull and aggressive. After witnessing a murder and calling the cops, Frank responds to the officers questions about the killer's description with a confrontational 'say, what's the deal here?' The dialogue felt a bit too punchy, too self consciously 'noir.' Luckily, we don't have to wait too long for Ann Sheridan to appear, because the film takes a big turn for the better once she arrives.
As the cops question Eleanor about her runaway husband, they discover a few oddities. She has no pictures of him, knows nothing of his friends, if he has any, doesn't seem the least bit interested in his activities outside of the house, and is completely dumbfounded to learn he has a serious heart condition that he's been taking medication for(she had assumed the pills were vitamins). Clearly things have not been happy in their home for awhile, and Eleanor is as much convinced that Frank has gone into hiding to escape their marriage as to escape a gangland execution. But still she won't help the police, and makes her own attempts to track down her husband, with the help of a friendly reporter who offers payment for an interview before Frank disappears for good.
Woman on the Run is, like most film noir, an allegory, but unusually it's an allegory for marriage, and how people so close can become surly strangers without even noticing it. Eleanor's journey to find her husband before the police or assassins do becomes a journey into her husband's secret life, or at least a life that was secret to her. She discovers his medical condition, his friends, the stories he's never shared with her, and the fact that he is still very much in love with her. It's powerful stuff, undercut, however, by the fact that Eleanor does not get a similar treatment. We never get to see her point of view, the slights that she has experienced in the marriage. We see only that she's misunderstood her secretive but loving and devoted husband, and that places the onus of this once-failing relationship almost entirely on her shoulders.
Still, the film isn't intentionally misogynistic, and by placing a strong capable woman at the center of the film it's actually pretty progressive. It's unfair to judge this film by our modern standards, and should only be judged against the measure of it's own time. In that regard, this film is quite enjoyable. Take in everything I've said so far, and then add in some great San Francisco scene work, a story with a compelling forward momentum, and an honestly exciting, literal rollercoaster ride of an ending, and the resulting film emerges as a hidden gem of the postwar noir movement.