[Update]
More recently, there is Amazon AWS ECR (Elastic Container Registry), which provides a Docker image registry to which you can control access by means of the AWS IAM access management service. ECR can also run a CVE (vulnerabilities) check on your image when you push it.
Once you create your ECR, and obtain the "URL" you can push and pull as required, subject to the permissions you create: hence making it private or public as you wish.
Pricing is by amount of data stored, and data transfer costs.
[Original answer]
If you do not want to use the Docker Hub itself, you can host your own Docker repository under Artifactory by JFrog:
https://www.jfrog.com/confluence/display/RTF/Docker+Repositories
which will then run on your own server(s).
Other hosting suppliers are available, eg CoreOS:
http://www.theregister.co.uk/2014/10/30/coreos_enterprise_registry/
which bought quay.io