Sure, it's possible to host your own mail server, except that you need to host it in a place that isn't port-filtered (eg: not on most residential ISP connections), and if you want anyone to be able to receive your emails you need to make sure you're not blacklisted-by-default by the big spam filtering engines. (eg: DNS settings, but also monitoring successful delivery rates and working with spam filter operators to determine why your emails get flagged and trashed.)