I think something like that might be feasible in the future with things like web bundles[1] and content security policy[2]. In theory you could have a site loaded as a bundle and a CSP that blocks everything not loaded through the bundle, effectively cutting off all network access.

You'd probably also need some sort of Feature-Policy[3] that prohibits access to any form of persistent storage so sites can't just save data until the next non-CSP-protected page load and transmit it then.

[1]: https://wicg.github.io/webpackage/draft-yasskin-wpack-bundle...

[2]: https://developer.mozilla.org/en-US/docs/Web/HTTP/CSP

[3]: https://developer.mozilla.org/en-US/docs/Web/HTTP/Headers/Fe...