Storing on your own hardware will always be cheaper (Backblaze has a great blog post on explaining why they built out their own data storage nodes at rented colo space because of this).
You don't have to "design for it". The default storage class for S3 is your data is automstically copied across three data centers. You have to explicitly specify "reduced redundancy". Yes you pay for it, but you don't have to do anything special.
I purposefully didn't use Amazon's wording because it would be confusing to someone who doesn't know about AWS.
An "availability zone" is an isolated data center. A "region" is a group of availability zones that are geographically isolated but somewhat close to each other.
For instance, three availability zones (data centers) that are within 100 miles (making up a distance) would make up a region.
https://www.backblaze.com/blog/petabytes-on-a-budget-how-to-...
https://www.backblaze.com/blog/wp-content/uploads/2009/08/co... (Cost of a Petabyte by service vs DIY)