Provided are a computer program product, system, and method for accessing records of a backup file in a network storage. A backup file comprises a sequential file of metadata records and data set records is processed. The metadata records include metadata on data of data sets in the data set records. Backup objects store the metadata and the data set records in the backup file. A determination is made of at least one container in the network storage to store the backup objects. Backup object information indicates for each of the backup objects, an order of the metadata or data set record, stored in the backup object, in the backup file, and a container of the at least one container in which the backup object is stored. Multiple of the backup objects are concurrently transmitted to the network storage to concurrently write to the at least one container.