No, it is not possible to open ArcGIS Explorer map files (NMF) in ArcMap. These files are exclusive to ArcGIS Explorer. Currently, there is no tool in Esri's core software that allows opening or converting NMF files to a readable format in ArcMap.
As an alternative, ArcGIS Explorer builds 2500 and 2505 provide the ability to share or export notes as layer packages (.lpk). A layer package includes both the layer properties and the dataset referenced by the layer. With a layer package, everything about the layer — its symbolization, labeling, field properties, and the data, can be saved and shared. Layer packages can be added to and worked with in ArcGIS for Desktop.
For more information, refer to the following Esri blog page: Sharing notes as layer packages.