I'm using 10.2. Our server is on a different domain from the one I'm working on regularly. If I connect to the server using a remote desktop session, I can create a database connection. We have a dedicated domain for our server at https://arcgis.domain.org/arcgis/rest/services. I have not been able to connect to the server through my local machine. Of course it won't work by just entering an instance. I've tried various URLs, and I've tried reading about this, but I'm wondering if it's even possible. Anything I've tried, I've received the "Unable to connect to database server to retrieve database list..." message.
Anyone know how or if this would work? I'd appreciate advice or suggestions.
It's unclear from your posting if you're asking about ArcGIS Server or enterprise geodatabase connection (or both). A graphical representation of the networks and nodes and software suites would go a long way to explaining the problem (and will likely provide the answer).
Generally speaking, you need full network connectivity in order to create a connection. I connect to both database and ArcGIS server nodes in my office through VPN all the time, but I could not, for example, use my Linux box to connect through my Win7 laptop's PuTTY session to remote nodes without making significant modifications to various firewalls and custom-coding more network proxies than I'd like to -- Just because you can get to two nodes from a location doesn't mean they can talk to each other.