I don't believe that the book explicitly states a year that the story takes place. In reading it, there was a feeling of a previous time period, but I think that was just because of the Old World type setting in the apartment building. As a modern American, living in a building with a Concierge probably feels old-fashioned.
But, in spite of that, I believe that it is meant to be present-day Paris. That is what lends it its charm - that these modern people still keep alive the older philosophies and literature. Human connection is universal and not really rooted in any particular time.