When Zotero can't access the network, it's usually caused by incorrect proxy settings or security software on your computer.
Zotero uses your system proxy settings by default, so disable your system proxy settings if you're not using a proxy or correct them if you are. Note that other software on your computer may not be using the system proxy settings, which is why other programs may still be able to connect to the Internet when Zotero cannot.
If you need to configure the Zotero proxy settings separately from the system settings, you can access the Config Editor from the Advanced pane of the Zotero preferences, apply the same settings that you would in Firefox, and restart Zotero, but the default setting (network.proxy.type = 5, to use the system proxy settings) is recommended.
If changing proxy settings doesn't help, try temporarily disabling any security/firewall software on your system.
Some connection errors can also be caused by certificate issues on your network.