As Nic mentioned, you can use gmail:
https://productforums.google.com/forum/#!topic/gmail/tEaJstfhzeIYou will need to prove to Google you own the domain, so you'll have to have a physical website at least temporarily. I use Hostgator, which is slow, but cheap/reliable ($8/month for single domain). I don't like Goddaddy, but they are cheap and good bandwidth (about the same price). If you're technically inclined,
https://www.gandi.net/hosting/ has $4/month but it's nonstandard - I've only read about it, I've never used it. Unfortunately, you can't trust online reviews - it's too easy for someone to just say 'it's great' so as to get affiliate income...
Another alternative is free hosting - do a search on 'free hosting' and check them out. Many display ads, which won't be an issue for you. With a quick review, I found two that looked interesting:
http://www.biz.nf/ and
http://www.freehostingeu.com/Hope this helps.