Now I was on fast-attack submarines (versus ballistic missile boats), but we regularly got email roughly every day when at sea. During certain operations, we would even regularly get an abridged NYTimes PDF news summary sent to us. And my grandmother died when I was underway on deployment - I got notified immediately. (Though obviously wasn't able to travel back.)
One important thing is that the IT admins on the boat had various scripts they'd run against all the email sent/received to help ensure that classified information didn't leak out, or troubling information (I think like family deaths?) leak in. But it never felt invasive.
Maybe the French Navy operates very differently than the American Navy, but... I call BS on this article.