Depends on what your company requires for restores. I think that's the big mistake a lot of IT people make - they assume they know these answers and they really don't. Only the business (IE your bosses or the stake holders in the company) knows how quickly restores are required.
For backups, if your business requires 15 minutes return on a backup, using cloud solutions isn't an option for you. If however your company doesn't care how long it takes to restore the data - then cloud works great. Generally most companies fall somewhere in the middle - where file servers and some application data can reside in the cloud because restore times will be hours if not days depending on the plan you have. While other applications (Tier1 applications) like Exchange, document management (if this is important to you) and other financial applications the restore time will need to be under an hour if not sooner.
You really need to break out all of the data you are backing up and find out for each piece of data (IE a piece of data being Exchange, File Servers, DFS, Application Data, Financial Data, ect) what the business expects for restores. Weigh what the company needs with costs and what your cloud provider can provide.
As for Disaster Recovery it depends on your current infrastructure. A lot of companies are going to Amazon, Azure and the like. We are toying with this right now but the big issue is cost and the ability to 'fail back' from DR back to your environment. The big issue right now that we are finding in DR is that you can easily fail over your Exchange or SQL environment to a node in the cloud - but what do you do once you've restored service on-prem? Putting those services back on site is difficult and depending on your provider may not actually be possible. Cost is another issue - running services in the cloud is cheap for a few servers using shared resources (IE your application server is sharing resources with a half dozen other companies running applications on that same server). So if you start requiring certain performance it gets really expensive. It becomes astronomical when you start looking at dedicated resources (IE paying Microsoft to run a server for you and you just manage the virtual machines on it) - dedicated servers are going to be required for things like HIPPA compliance and other regulatory items.
