Let me be absolutely blunt about this:
The cloud doesn't scare me.
To start, the cloud isn't for everyone. Plenty of businesses – across plenty of verticals, and in plenty of countries – have all kinds of restrictions that prevent them from moving their SQL Server estates into someone else's data centers. While more companies are gradually starting to dip their toes into the cloud, this isn't happening overnight; for a substantial portion of the market, it might never happen at all.
I am less a DBA than I was 10 years ago, and to be honest – even for the folks that are hardcore DBAs today, and whose employers have bought into the cloud – the cloud isn't taking your jobs. I think the biggest parts of instance management that the cloud absolves you from have been gradually moving away from you already, starting years before the first time you heard the "Azure" brand. Storage shifted to SANs, clustering was handled by Windows admins, virtualization has its own set of experts, and even backups and patching were often taken off your hands (not always for the better).
There is still plenty to do. If your business thinks it can make you redundant by moving your databases to the cloud, it has another thing coming. You don't just fire up an Azure SQL Database and then look for something else to do. You still need capacity planning. You still need to deal with complex applications that span more than a single database, or the work required to consolidate those. You still need to handle design choices, application interfaces (e.g. stored procedures), index maintenance, query tuning, performance troubleshooting, and yes, even backups.
Yes, some of your options for high availability and disaster recovery have changed, and your response to performance issues will change as well. But don't be afraid of this. I've barely touched the surface here, but if the cloud makes you nervous, I do urge you to read Kendra Little's post. Embrace it, study it, and be prepared… the worst that could happen is that you might not need those skills yourself, in which case you can still use them to help others.