My current company spends a few hundred dollars a month on S3. I certainly couldn't match S3's uptime at that budget. Maybe I could do it at 2-3X the budget, but with that budget it might be easier to mirror my files across multiple S3 regions.

Exactly. Build multi-region support into your app(s), enable S3's replication so objects in your primary region are replicated to another region, and then properly handle loss of a region (go read-only or write to another region and restore consistency later).

You'll still be spending less than attempting to maintain a highly durable object store yourself across multiple datacenters or geographic areas.