A good backup system ALWAYS should make provisions to isolate the device carrying the data for safety, protecting it from electrical glitches, etc and physical damage. Among other things, that means you ought to be able to disconnect it completely from any electrical supply and from the computer. Ideally, the unit should be physically stored in a different location to guard against rare events like fire, flood, etc.
For these reasons, an external HDD makes much more sense, even if you don't store off-site.
You want an example of rare dangers? I heard a story that one professional data center operating in one of the World Trade Center towers did it all right, including storing backup tapes off site - in the other tower!
An online internal backup is not a backup. It will be stolen along with original data, zapped by the same power spike that kills the original data, corrupted by the same virus that corrupts the original data, killed by the same toddler that knocks over the original data, etc. etc. etc.
Good backup policy says that you should have two offline copies of your data, one of which is stored offsite. It only seems like overkill until it saves your bacon...
I personally like getting a separate enclosure and installing the drive myself. That way I know exactly what model drive I've got and I also know that if I want to upgrade the drive later it will be easy to do. Some of the pre-assembled enclosures really aren't conducive to drive swaps.
If you're planning to use USB to connect the drive then you might want to think about looking for a USB 3.0 enclosure and USB 3.0 adapter card for your computer to connect it to - it will probably be around twice as fast as a USB 2.0 enclosure. But if you're planning to use eSATA then it probably won't matter.
There advantage of a pre-made complete external drive are that you don't need to do any assembly and many of them come with decent automatic backup software included in the price. On the other hand, I'm like sminlal - I rather "roll my own" - so I bought an enclosure and chose my HDD and backup software tools the way I like them. I chose an enclosure with no cooling fan (quiet, and no fan to wear out) and a HDD with reported relatively low heat output. That works just fine. I chose an enclosure to accommodate a SATA II drive, and that had both USB2 (almost universal now) and eSATA (faster) interfaces to the computer. Cases for 3½" internal HDD's now almost ALWAYS come with their own power supplies (in the case or external "wall wart" or box) because the power required for the larger drive units is simply too much to draw from any USB port (and, of course, there is no power available from an eSATA port). Like many, mine has its own on/off switch and an indicator light on the front.
I built mine a few years ago an today porably would choose different components - not that mine's bad, but just there are really good choices now. I second the suggestion of WD Green line drives. They may be SLIGHTLY slower in terms of having to ramp up their speed when used after a no-use period, but that is completely acceptable in an external backup unit. They are great for low heat output and power consumption, and every other aspect of them is the same as the rest of the highly-reliable WD line.