I'm 100% certain Google lets you get Gmail for your domain. You have to pay for it (but you have to pay for FastMail), and you get other services alongside Gmail, but it certainly exists.
Using Google Apps (which is cheaper than FM)
I pay $15 per year for a FastMail family account,[1] which allows me to use my own domain. The account includes only 200MB of space, but that works for me because I delete email from the server so that the US government can't access it, which they can do without a warrant.[2][1] https://www.fastmail.fm/signup/family.html
[2] https://www.eff.org/deeplinks/2012/11/when-will-our-email-be...
You're still able to get a free one-user Google Apps for Domains account by going through Google App Engine. [1] I've used it to get catchall email addresses for any new domains I've recently registered.
[1] http://lifehacker.com/5967336/use-google-app-engine-to-get-g...
But Gmail/Google's ease of use is its USP. You can have chat in browser, GTalk, Adium, Android , Pidgin and all then synced to mail box.
You can have calendar everywhere, taks..you name it.
So, until some business - like FastMail, PoBox, come up with such close integration I don't think there's much sense to move other than privacy and paranoia. Because once outside you end up setting up a workflow dependent on very different services and worry which one is gonna break when and then look for sth else.
1) A black navigation bar across the top that doesn't help with email.
2) 300px wide box across the top right dedicated to a social network no one uses (not email related).
3) A loading screen when the page first loads.
4) New mail compose window (no need for an explanation here).
5) Ads