In the ASP.NET world, RavenDB is a popular option. For those of you brave enough to take the NoSQL plunge, I wanted to guide you through getting started and mention a few things. RavenDB can run in many modes.
The one we are concerned with when it comes to running it on a shared hosting account is Embedded Mode. This essentially embeds the RavenDB goodness into your application and will work on a shared hosting account. In fact, there is even a nuget package that you can use to drop it into your project.
Embedded mode even contains an HTTP server. However, this option will not work in a shared hosting account. So do not set:
UseEmbeddedHttpServer = true
Your best option would probably be to add something like this to your code:
#if DEBUG UseEmbeddedHttpServer = true; #endif
That way you could change your build type for either working locally or for production when you publish. For those looking for a fantastic tutorial and sample application, look no further than here.