SpiderOak does not need any more network access than a typical web browser. In unusually restrictive networks, SpiderOak may need to be given permissions to make an SSL connection to spideroak.com and *.spideroak.com port 443.
Note that SpiderOak also uses SSL and careful certificate verification for all network connections. Because of this, SSL proxies that attempt to monitor traffic with an intermediate certificate will cause SpiderOak to fail. The SSL proxy must accept the CONNECT method. Also, SpiderOak's SSL certificate verification does not rely on any certificate signing authority, and is thus resistant to DNS attacks.
Remember, also, that your data is encrypted before ever leaving your system, so your data is secure on both private and public networks.