Based on the updated information in your question, I think it is pretty clear that you need to have a license of ArcGIS for Server. This software now has ArcSDE integrated into it.
Based on this article: Storage in a Relational Database Management System - I think it is clear that to create an ESRI Geodatabase in PostgreSQL, which is an RDBMS, you need the ArcSDE technology.
This document shows the breakdown of the Multi-User Geodatabase types offered by ESRI.
The only other types of ESRI Geodatabases, are the Single User Geodatabases. There are two types, the Personal Geodatabase and the File Geodatabase. These can both be created using a standard ArcGIS installation, or you can use the OGR libraries to create them outside of the ESRI software products, using the File Geodatabase API.
Update for clarity
If you are using PostgreSQL, or another supported RDBMS as your spatial data repository, there are two ways you may access data.
- If you wish to see ESRI object types like
Raster Catalog, etc; or perform ESRI-based operations like
synchronization, then you must go through this installation process: ArcSDE 10 for PostgreSQL Installation Guide This process requires the ArcGIS Server license that you have.
If you solely want to store geographic data in the RDBMS using the Open Geospatial Consortium's Implementation Specification for Geographic Information, Simple Features Access for SQL, which is an open standard, and access it through ArcGIS, then you need to do the following:
- Install PostGIS for PostgreSQL - This provides the geographic object support for the RDBMS.
- In ArcGIS, you will use the Make Query Table command to create a spatial layer or data table using an SQL query from the PostGIS spatial database.
This does not require a license of ArcGIS Server, but again, your data will not be stored in the ESRI object types listed above.