I disconnected my notebook from the network and Darteditor runs just fine. I can start web pages without any problem.
Probably Darteditor tries to download pub dependencies which won't work without a network connection.
If you can open a public website like google.com in your browser then it's probably not a firewall limitation. AFAIK Dart uses only HTTP port 80 which is usually open. If your HTTP connection goes over a proxy that might not work - that is a common problem with Dart.