This is quite late but I just watched the movie this morning during TCM's 31 Days of Oscar. To answer your question:
Paula is holding the pole part of what's known as a Police Lock. The top part fits into a 'lock box' on the back of the door while the bottom of the pole usually fitted into a small slot that was positioned into the floor made of metal. Supposedly it helped 'hold' the door if someone was trying to push in on it. Some very old apartment buildings have these still around the city, mostly in the oldest areas. At this point in time, many still have the floor slot (you can tell the age of the building this way) while the lock part or the poles could be long gone. Occasionally they are still intact.
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