NearlyFreeSpeech looks good. It runs on a pay-as-you-go model and shouldn't cost more than the initial deposit if the site is small enough. It looks like tech support is a premium for non-critical issues, though.
I hate Google sites and Google functions except GMail and Maps. Their docs function never worked properly for me (and is SLOOOOW), Groups were unreliable and always experienced downtime, ugh. It's just me though, maybe your milage may vary.