1. A method, comprising: performing, by a storage service implemented on one or more computer systems, wherein the storage service stores a plurality of objects for a plurality of users and wherein at least some of the plurality of objects further comprise object data: receiving, via an interface of the storage service, a request for a user of the plurality of users to enable or suspend a versioning state of one or more objects stored by the storage service for the user; and enabling or suspending the versioning state of the one or more objects stored by the storage service for the user in response to receiving the request, wherein: when the versioning state of a particular object of the one or more objects is enabled, a new version of the particular object is stored by the storage service in response to one or more operations performed on current object data for the particular object, and when the versioning state of a particular object of the one or more objects is suspended, a new version of the particular object is not stored by the storage service in response to the one or more operations performed on current object data for the particular object.