As a start, be present in multiple EC2 availability zones (not just US-east-2, basically) and regions (this is harder). Cross-region presence needn't be active-active, just a few read-only database slaves and some machines to handle SSL termination ("points of presence") for your customers on the east coast. Perform regular "fire drills" where you actually fail over live traffic and primary databases from one AZ/one region to another.

"Building your own" is also something very few people (including Amazon itself up until fairly late, probably after the IPO) do: you can use a managed hosting provider (very common, usually cheaper than EC2) or lease colo space (which doesn't imply maintaining on-site personnel in the leased space: most colos provide "remote hands"). You can still use EC2 for async processing and offline computation, S3 for blob storage, etc... or even S3 for "points of presence" on different US coasts, Asia/Pacific, Europe, but run databases, et al in a leased colo or a managed hosting provider.

Yes, these options are more expensive than running a few instances in a single EC2 AZ: but that's the price of offering high availability SLA to your customers. It's a business decision.