Put up a RPI at home. postfix + dovecot + roundcube should do the trick.
Add FW forwardings for 80/443/25 (or allow IPv6 to pass through)
Update DNS records every N minutes. (cron, nsupdate, dyndns clients, amazon command line tools....). You will need SPF/DKIM.
The RFC for SMTP says Mail Servers have to retry for 7 days before giving up on mail delivery. This should be plenty for your home server. There are also commercial Mail relay and backup MX services (sometimes even as a free offer for buying domains on website X).
You can backup the SD-Card whenever you want. Your Mail stays at locations you control.
I currently have a root server, but I'm heavily considering "in-housing" those services because of the NSA activities.
EDIT: it's 7 days PS: Some old firewalls block dynamic IPs for mail delivery. I'm not sure how common this is today, especially as SPAM and botnets have evolved a lot.