"Server unreachable" is actually a poorly-worded error that appears to be associated with trying to log in with an unconfirmed account.
When you first create the account, Crytek will send you an email to confirm the account. Until you actually confirm the account, attempting to log in through the CryENGINE Launcher will give you the error "Server unreachable".
Unfortunately, confirming your account may also be bugged.
Sometimes, confirming your account may be harder than you would think. Crytek send you the email, and you click on the link to confirm the account. This takes you to a page that acknowledges your confirmation, but the launcher still gives you the "server unreachable" error.
Sometimes, the link takes you to a page asking you to log in. When you attempt to log in, you are asked to confirm your account. If you follow the link provided via email, it takes you back to the same page, and the vicious cycle repeats itself. Attempting to resend the confirmation email may, instead, send you a thank you for confirming your email. Despite this apparent acknowledgement, you still can not log in to the CryENGINE Launcher.
This appears to be a known, yet uncommon, problem. Unfortunately, the only direct solution I have found is to contact Crytek, directly. The only times I have seen this solved is through direct communication on the forums,
which in turn requires you to have a confirmed account to access, giving you a painful Catch-22 not being able to contact Crytek was especially relevant to my question and answer, but I have since found contact through email at firstname.lastname@example.org. This may be a helpful point of contact, if anybody runs into the same problem, and really wants to keep their original username or associated email.
By far, the easiest solution is to make a new account with a different email.
After several weeks of frustration, this solved the problem, for me. If you signed up with a particular email or username you really must have, use your new account to communicate with Crytek, who can directly toggle the "confirmed" switch on your original account.