Some providers just run big pipes into the back of their servers and run the data-center network as a slightly more secure Internet. Your data can stay separate from that of other customers if it's encrypted in a virtual private network (VPN) tunnel, but the packets that contain the encrypted data are bumping along through the network wires, routers and switches along with packets belonging to other companies.
Some providers will create more isolation for your cloud by providing either separate physical connections to the Internet – which requires at least one separate network interface card on each server dedicated to your traffic alone. If you don't share physical servers it's a little easier, but you'll still end up paying for separate network connections and routers or switches that provide a private channel to the 'net or through a private (MPLS) connection to your data center.
A few providers are able to virtualize the networks as well, dividing 1GB/sec or 10GB/sec network pipes into separate streams and dedicating one or more to your data.
Virtual I/O setups can also isolate network traffic, but isn't as common as some of the other methods.
Storage in a cloud facility will be virtualized, so you can't usually point to a specific disk array as "yours." How separate is the storage? Is all your data on a completely separate array? Is it isolated on a different SAN than other data? If you share the same SAN, how are the data keps separate and what is the chance of you being damaged if another customer does something irresponsible – like getting raided by the FBI, which comes to confiscate the boxes with their data? Do your data go in for interrogation, too?
Are all the data sets encrypted separately and identified in ways that make it clear to anyone examining the storage that your data is just an innocent bystander and not implicated somehow?
Otherwise the Feds might feel they have a right to sift through your very personal corporate information without paying much attention to the security policies you assembled and delineated before putting it in the cloud in the first place.
Most cloud providers would replicate your data to another SAN or array so if one box goes down (or is taken downtown) you'll still have access to it. Does yours?
If so, how up-to-date is the data? You probably don't want to put highly time-sensitive transactional apps up in the cloud, so losing a few seconds or minutes of data might not be a big deal. What's the window, though? If someone trips over a power cord, even, how much data are you likely to lose?