Running the ArcGIS Data Store describedatastore utility returns the following error message:
Error encountered: No valid connection to ArcGIS Data Store configuration store established.
Caused by: Connection to localhost:9876 refused. Check that the hostname and port are correct and that the postmaster is accepting TCP/IP connections.
There is no postmaster.pid file and the ArcGIS Data Store Windows service account has sufficient permissions to modify or delete a .pid file. So, the issue is not the one discussed in the below article.
Some things you can check
In the Task Manager on the server you should be able to see "PostgreSQL Server" is running. In the "details" tab you should see about 20 copies of "postgres.exe" running (as the user you picked when you set it up)
Can you run a netstat command (in a command window) on the server and look for the open ports? Try
netstat -an
and look for lines similar to this
TCP [127.0.0.1]:9876 [0.0.0.0]:0 LISTENING
You might also see a line
TCP [0.0.0.0]:9876 [0.0.0.0]:0 LISTENING
This means it's listening on all ports. You should see the other line 127.0.0.1 too.
If you do NOT see that it means the service Postgres is not listening. Probably because it's not starting.
You can skip all that and go straight to the Postgres log file, which is found in different places if you are running Linux or Windows. Based on the error page you mention I assume Windows. On my machine, logs are in C:\arcgisdatastore\logs\NAME where "NAME" is the name of the machine. In there you will see folders for "server" and "database". Depending on what's wrong you will find messages in one of those places.
"server" is the controller, that runs postgres database, that is "arcgisdatastore". "database" is postgres. When things start up you will see messages in the server log about it trying to talk to postgres. In the database log you will see actual messages from the database. If it's running.