It sounds like you have not set up port forwarding on your router to forward requests to 5003 on your router's IP address to the machine hosting your database. The steps for setting up port forwarding vary with the internet router and you'll need to determine whether your router's IP address is static or not. (Usually it isn't unless you get your ISP to give you a static IP address).
I do have that set up 5003 to 5003 to machine that is serving
Re: Could not see filemakes files on server when connecting remotely and got error "Either the host is not available, or the file is not available on that host."
Turned out that my problem (Windows 7 64-bit, Filemaker 9 and 11) was that Apple Bonjour was either not installed or had been uninstalled. A reinstallation of Filemaker on the remote laptop fixed it.