I am looking for a tool (Python library etc.) that would allow me to edit a file geodatabase without Esri software?
It is possible?
You could utilize my Python wrapper for the File Geodatabase API.
Read about it here: http://libjoe.blogspot.com/2014/02/python-wrapper-for-esri-file.html
I also started an open source project for it: https://code.google.com/p/file-geodatabase-api-python-wrapper/
As wrote before + The File Geodatabase must be created in ArcGIS 10(+). File GDB's created using ArcGIS 9.X are NOT supported.
Also it is possible with Qgis with the mentioned library - well I only opened the file, didn't try to edit - more here: File Geodatabase (*.gdb) support in QGIS?
(I wanted to wrote this as comment to previous answer but I don't see any link "add comment")