This is loaded, but very simple if you do not trust them.
I have had Yahoo, Hotmail, Aol, Lycos, Excite, accounts for over 10 years and Gmail since it came out, I have never heard any problems you say, they are backed up and backed up constantly to several different servers in case of a problem, they scan your emails for virii and malware and such, so on one hand your "untrustworthy" statements bothers me and makes me laugh. BUT since "it is not in your hands" that makes you weary. So the only way to save them for your self is to use an Email Client ( a local (installed on your computer) email program like Outlook, Outlook Express, Thunderbird or similar) What happens though is good and bad, they automatically download those emails to your computer if something is infected it may infect your computer, there are plenty of settings though, some you can download a copy leaving copies on the free email or not and stuff like that, if your computer is infected it can infect those emails and or your email client, and another possibility is you can lose all your emails on your computer if you get infected and the computer dies, why not leave them out there with TRUSTWORTHY places like those mentioned where they are fairly safe. Again you can do both, Have copies.
I do not suggest Firefox Addon's or extensions those DO have problems and you might lose ALL your emails the very first time you try to connect to yahoo or gmail or hotmail. Make sure you use a separate program/client.
The reason people stopped using email clients like Outlook and Express and stuff is because emails are downloaded to your computer, so they then had to try and make them safer by adding settings that some times work sometimes don't. Like previewing.
Millions and Billions of people use and have been using Free Online Email day in day out year after year.
If you want FULL POWER, get a client download "copies" to your computer hopefully a external drive in case your computer ever fails or burn them to CD or DVD and make copies of them too.....