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Storing backups in Azure Blob storage that utilizes the archive tier is the most cost-effective option for minimizing expenses while meeting compliance requirements for file-based database backups that are rarely accessed. The archive tier is specifically designed for data that is infrequently accessed and can sustain longer retrieval times. With significantly lower storage costs compared to other tiers, this option is ideal for compliance scenarios where data needs to be preserved but is not actively used.

Using Azure Blob storage in the archive tier is particularly advantageous when you need to retain large volumes of data, as the reduction in costs allows organizations to allocate resources more efficiently. Additionally, retrieval times for data in the archive tier can exceed 24 hours, making it suitable for backups that do not require immediate access.

Other storage options, like Azure SQL Database or Azure Storage with the Cool tier, are more expensive and are designed for different use cases where data access needs to be more frequent or immediate. An Azure Recovery Services vault is primarily tailored for backing up and recovering VM workloads and applications, rather than for long-term storage of infrequently accessed backups. Therefore, using the archive tier of Azure Blob storage strikes the right balance between compliance needs and cost efficiency for rarely used backups.