Final answer:
The cheapest S3 storage classes for long-term retention of rarely accessed important data are Amazon S3 Glacier and Amazon S3 Glacier Deep Archive, with the latter being the most cost-effective for very long-term storage.
Step-by-step explanation:
The cheapest S3 storage class for important data which needs to be retained for long periods and is rarely accessed is the Amazon S3 Glacier and Amazon S3 Glacier Deep Archive. Amazon S3 Glacier is designed for data that is accessed less frequently, but still requires rapid access when needed, whereas S3 Glacier Deep Archive is meant for data accessed extremely infrequently and is the most cost-effective option for long-term storage.
Data stored in these classes is generally not needed for months, years, or even decades, and can be retrieved within minutes to hours using S3 Glacier, or hours to days with S3 Glacier Deep Archive. The pricing is significantly lower than other S3 storage classes, but data retrieval may incur additional costs and take longer compared to more readily accessible classes.