Off-site remote backup using a network can be a great idea if you have the right circumstances for it, otherwise you're spending a TON of money to try and make something work that just simply won't.
At our office we once considered putting in a simple backup server at our other office (located an hour and a half away) that would back up all the data from our primary office remotely there in case of a disaster. Here's why we decided against this though:
1) Cost: To set this up, we'd be needing to add another server of some kind, even if it weren't incredibly powerful, that could store at least a terabyte of data in a RAID array just in case of failures, and would have to be able to reside on the network and accessible from the outside world while still remaining secure from viruses, malware, intrusion, etc. etc. Basically, the best way to do this is to get a dedicated firewall appliance with a VPN tunnel capability. The cheapest device that I know of that can do this is the SonicWall TZ100, and we would have to have one for each side of the network to build the tunnel and secure both locations (the source file server in Colby, and the destination backup server in Hays.) On hardware alone, you're looking at an initial cost of more than $1,000.
2) Network Connections: In rural America there aren't a whole lot of great internet speed offerings yet. We're using the best DSL package available at our Colby office of 3Mb/s download 1Mb/s upload, and at our Hays office have been running 1.5Mb/s download and 300Kb/s upload. This speed is WAY too slow to be transferring large chunks of data across the network from one place to another on a regular basis. We estimated that on average our shop has about five to ten gigabytes of data that is created or modified in some way during the course of a day, so if we have the server set to automatically back up nightly all new data (again, an incremental backup of JUST new and modified data) it would take approximately 12 hours or more to transfer that data across the network, and you better hope that you don't have a network drop during that time or you can corrupt your data and your backup fails. And networks around here drop regularly... My internet provider is known for having at least one network drop per day.
Given the massive cost to perform this backup, it's much more feasible for us to make remote backups using external hard drive to an off-site secure storage like your bank or another safe. Instead of $1,000 up front costs and having to pay more than $100 per month per location for a very slight increase in internet speed, it's more feasible to buy a few high-capacity hard drives and do the backups yourself, even if you have it set up on a schedule to run automatically and you just move the drives off-site. If internet speeds around here were faster, less expensive, and the data that we had to push across the network was less, then yes off-site network-based backups would be great to use but only if you have control of the backup as well.
Just remember, DropBox has been under fire lately since announcing in their privacy policy that they in fact do have access to all data on their servers. It's only company policy that technicians should not access the data, but that doesn't actually prevent them at all from having the ability to do so. I believe that cloud computing can be great in the right circumstances and for the right reasons, but it is not to a point of being universal by any means. It's a tool to perform a specific job, just like you wouldn't want to try and cut through a sheet of plywood using a pair of scissors.
.