In contrast to the currently accepted answer, currently in 2024 the encryption of data at rest for Azure Service Bus is limited to the Premium tier only:
Azure Service Bus Premium provides encryption of data at rest with Azure Storage Service Encryption (Azure SSE). Service Bus Premium uses Azure Storage to store the data. All the data that's stored with Azure Storage is encrypted using Microsoft-managed keys.
https://learn.microsoft.com/en-us/azure/service-bus-messaging/service-bus-premium-messaging
Also referenced in this [Microsoft Q&A][1]:
@Antra Sharma Azure service bus REST Encryption are only supported in
Premium namespace and it cannot be enabled for standard tier Service
Bus namespaces. As premium namespace uses Azure Storage to store the
data therefor it provides encryption data at rest with Azure Storage
Service Encryption (Azure SSE).
[1]: https://learn.microsoft.com/en-us/answers/questions/424372/azure-service-bus-encryption-at-rest-transit-(stan