A technique for managing data objects in a data storage cluster realizes an application-accessible data object within a file that belongs to an internal file system. The internal file system runs on a computing node of the cluster and is backed by a LUN realized in block storage. The storage cluster achieves snapshots of the data object at the level of the file system, e.g., by snapshotting the file, such that the data object and its snapshots are all backed by the same LUN in the block storage.