In ArcGIS Desktop we are able to create a file geodatabase and also a feature class.
Is this possible in QGIS?
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As far as I know there is no way to create a file geodatabese (esri format) with qgis Edit : see comment below.
You may only read or read/write a file geodatabase (depending on what driver you have as said in the comment below your question)
The closest approximation of a file geodatabe in qgis would be to use GeoPackage or Spatialite (the answer to this question give more detail on these format). If you need to keep compatibility between qgis and arcgis these two format are supported in arcmap from version 10.2