I use Amazon AWS S3 for my Muse site. It seems really complicated at first but it really isn't that bad. Super super cheap too... like, pennies. More info here: http://alignedleft.com/resources/cheap-web-hosting (That page takes a look at a bunch of other hosting options as well.)
If you want something much easier and free but with less control, you can use Dropbox with Pancake.io and it will even update automatically when you export Muse to your Dropbox folder. Only downside for me was that I wasn't able to change my site's favicon from the Pancake.io default. They didn't reply to any of my support inquiries, so I'm not sure what's up with them.
Let me know if you have any questions.