To do that you basically you need to forgo all the benefits of cloud computing and think of it like hosting at two traditional datacentres.
It's not unsolvable, but as others have mentioned it's probably not worth the time or money needed to have a live standby. There will be a level of failover speed that is worth having, but that might be "if EC2 is gone we can recover in 48 hours with no more than 6 hours of data missing" so the contingency will not kick in for a short EC2 outage.
I fail to see how this is related to "forego the benefits of cloud computing"... Does your startup have the resources to manage multiple datacenters by itself (such as one in US and one in EU)? Isn't this a clear benefit of cloud computing (and a pretty big one too)?
I should have been clearer - you forgo benefits of cloud computing for the functionality related to having a second cloud installation to failover. Each cloud will have all the benefits, but between them you're needing to come up with a way to synchronize applications and data that isn't just "create a cloud backed relational DB and point all the app servers to it"
It's not unsolvable, but as others have mentioned it's probably not worth the time or money needed to have a live standby. There will be a level of failover speed that is worth having, but that might be "if EC2 is gone we can recover in 48 hours with no more than 6 hours of data missing" so the contingency will not kick in for a short EC2 outage.