1. A computer-implemented method for providing access to private computer networks, the method comprising: receiving, by a configurable network service implemented by one or more configured computing systems, configuration information including user-specified network topology information for a private network extension to a remote private computer network, multiple user-specified network addresses for use by the private network extension, and an indication of a remote resource service that is external to the remote private computer network and external to the private network extension and that is separated from the remote private computer network and from the private network extension by one or more intervening networks; creating, by the configurable network service, the private network extension in accordance with the received configuration information, the creating including configuring the private network extension to provide private access between the private network extension and the remote private computer network in accordance with the specified network topology information; generating, by the configurable network service, a unique identifier for the private network extension to represent a first namespace within the remote resource service, the first namespace to include one or more computing-related resources provided by the remote resource service that are accessible only from the private network extension; configuring, by the configurable network service, a local access mechanism that is part of the private network extension and that enables the private network extension to access the one or more computing-related resources within the first namespace, the configuring of the local access mechanism including assigning one of the multiple user-specified network addresses to represent the remote resource service and associating the unique identifier with the assigned network address, so that communications sent from the remote private computer network or from the private network extension to the assigned network address are modified to include the unique identifier to enable the remote resource service to identify the first namespace and are forwarded to the remote resource service over the one or more intervening networks; and providing, by the configurable network service, the private access between the private network extension and the remote private computer network.