Well, cloud-based disaster recovery means a lot of different things. Some of them are pretty similar to what people use in traditional active DR environments and others are pretty unique to cloud. So, you know, for one thing, you could just park data out in a distant repository which is not that different from driving a tape to an Iron Mountain facility. Instead of actually driving the tape there, typically, you are sending the data across a wide area network (WAN). But once it's there, it's just a repository and it's available to be restored in the event of a failure at the primary site.
These days, there are also a number of cloud DR services which allow users to replicate instances of virtual machines to the cloud. These solutions place compute resources in the cloud alongside data in cloud storage and allow users to run critical applications in the cloud while the primary site is restored. This of course is very different from just parking data on tape in an Iron Mountain vault as it allows for application availability during restores.
As far as cost goes, it's so hard to give a blanket answer for that. It really depends on what it is that you're building. Cloud is relatively cheap to park data that you are never going to use. But, tape storage offsite is pretty cheap in up in itself. It depends how many redundant copies and the type of cloud DR you choose. Those are key factors to consider when determining which one would be cheaper. So, it really depends.
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