Understanding the Requirement: The application running in a private subnet needs to store and retrieve data from S3 without data traveling over the public internet.

Analysis of Options:

NAT Gateway: Allows private subnets to access the internet but incurs additional costs and still routes traffic through the public internet.

AWS Storage Gateway: Provides hybrid cloud storage solutions but is not the most cost-effective for direct S3 access from within the VPC.

S3 Interface Endpoint: Provides private access to S3 but is generally used for specific use cases where more granular control is required, which might be overkill and more expensive.

S3 Gateway Endpoint: Provides private, cost-effective access to S3 from within the VPC without routing traffic through the public internet.

Best Solution:

S3 Gateway Endpoint: This option meets the requirements for secure, private access to S3 from a private subnet most cost-effectively.