I'm not usually one for scandals (in fact, I couldn't care less who government employees are sleeping with). However, it did bring up deep privacy concerns—how exactly did the government get a hold of the e-mails?
The EFF had released an article answering some questions about the scandal, which is worth a read. In particular, you should take a look at the EFF's Surveillance Self-Defense website for an in-depth summary of the laws surrounding government surveillance and tips on how to protect against it.
I'd like to touch upon a couple things. In particular, the article mentions:
Broadwell apparently accessed the emails from hotels and other locations, not her home. So the FBI cross-referenced the IP addresses of these Wi-Fi hotspots ¿against guest lists from other cities and hotels, looking for common names.¿To stay anonymous in this situation, one should consider using Tor to mask his/her IP address. Additionally, remove all cookies (or use your browser's privacy mode if it will disable storing and sending of cookies for you) and consider that your User Agent may be used to identify you, especially if maleware has inserted its own unique identifiers.
Also according to the EFF article:
According to reports, Patraeus and Broadwell adopted a technique of drafting emails, and reading them in the draft folder rather than sending them.That didn't work out so well. Consider encrypting important communications using GPG/PGP so that (a) the e-mail cannot be deciphered in transit and (b) the e-mail can only be read by the intended recipient. Of course, you are then at risk of being asked to divulge your password, so to avoid the situation entirely, it would be best to delete the e-mails after reading them. Additionally, if you host your own services, it may be wise to host your own e-mail (guides for doing this vary between operating system, but consider looking at software like Postfix for mail delivery and maybe Dovecot for retrieval).
Privacy isn't only for those individuals who are trying to be sneaky or cheat on their spouses. Feel free joining the EFF in trying to reform the ECPA to respect our privacy in this modern era; storing a document digitally shouldn't change its fundamental properties under the law.