In terms of MongoDB like others have said, you need the ability to install the MongoDB software and run it (normally as a daemon). However, hosted services are just beginning to appear, such as MongoHQ. Perhaps something like this might be appropriate once its out of beta (or if you request an invite).
It appears hosted CouchDB services are also popping up, such as couch.io or Cloudant. I personally have no experience with Couch so I can be less certain than with Mongo, but I'd imagine that again to run it yourself, you'd need to install the software (and thus require root access).
If you don't currently have a VPS or dedicated server (or the cloud-based versions of the aforementioned), perhaps moving your data out to a dedicated hosted service would be an ideal way to go to avoid the pain and expense of changing your hosting setup.