What's your understanding of "private domain"? Please define.
Let's say the private IP of your webssite is 192.168.24.35 and so you access the website in Google Chrome using the URL http://192.168.24.35. In this case you should be able to use the command:
ping 192.168.24.35
on the command line and get the echo replies displayed.
google.com is definitely not a private domain and if your PC cannot find the host you probably don't have internet access.
I'm looking for the ip of the server from a local network.
The IP of which server? If it's a local server we won't be able to tell you the IP. If you cannot ping it, you probably also won't be able to access it by HTTP (although theoretically it's possible but on local networks ping or icmp echo requests usually should work).