Many web hosting services like DrupalGardens.com and WordPress.com allow you to use your own domain. However, the set up procedure for many of them means that it is impossible to use an existing domain without some kind of downtime.
Contact us forms on websites allow the user to fill in the form, and the contents are emailed back to the site owner. I often see people put the address the user gives as the “From” field in the email – here is why you shouldn’t do that.
I did start this blog on wordpress.com. If your wanting a decent blog with reliable hosting where someone else takes care of all the details, that’s great. But when I tried to purchase a premium service to use a domain name on my blog, I ran into problems and ended up self-hosting.