Can anyone recommend a sturdy external hard drive specifically designed for travelling that I can back up photos and videos on?
Hard drives are not sturdy. Even ones with so-called rugged casings. What is much more sturdy and incredible durable is an SSD. This is a Solid State Disk with no moving part.
Compared to hard-drive, an SSD is more expensive, particularly a huge one. However for your purpose a smaller one will probably do an good ones under 240GB are quite reasonably priced.
You need to buy an enclosure for it. They are standard ones which are very cheap and self powered via USB. Those require a laptop or netbook to operate. They are also dedicated enclosures with flash-card readers built-in that are battery powered and copy media for you.
Personally, I have stopped using all such devices because they are also a target for thieves since they are costly. It is better but slower to burn optical disk which I do in duplicate. One copy is kept with me but separate from expensive gear while the other is mailed to myself. No such disk has ever disappeared. After all, it has no value to anyone else and cheap to mail in a small envelope.
Since use it for photography, I am assuming that you are carrying a camera and a laptop around with you too.
As mentioned by Itai, I would go for a rugged SSD drive. Everything else is a bit tricky if you plan to have a rougher trip.
What you need to consider however is that even a non-rugged drive will be in any case more stable than your laptop itself, since it does not have a screen. So you need to decide if you need something that will outlast the rest of your equipment in physical terms or simply something that serves as an emergency copy or an extended storage for your laptop.
So if you want to go rugged, LaCie has offered such disks since a long time, and now also has an SSD available. LaCie is however orange and might stick out again as a risk for theft. More easy to the eye is the ioSafe model.
If you can care for proper packaging just as for your laptop then any SSD will do it with a normal USB3 enclosure, without special rugged treatment. Such a SSD in a double layer ziplock to additionally protect against humidity is small and inconspicuous enough to be save from theft (unless you leave it lying around, but then in all likelihood your laptop will be gone, too) and at least as safe from damage as your laptop (which will be more expensive in any case). Padded packing just like for your laptop is required of course.
If you are most afraid that you might lose some images and absolutely want to diminish the risk of images getting lost, you might want to get a bunch of DVD Roms, burning them on the way and mailing them home as Itai described. Depending where you are going however, even that might not be the best option, since I can imagine that sending mail home from Papua or the Amazon Rainforest might not be a really good option either.
You can buy ruggedised drives, and they do have better shock resistance specifications. However, based on my reasonably extensive experience, you do not need anything exceptionally special if you take due care, a,d you can add an immense amount of shock resistance to a portable drive at almost no cost. The only thing you do add is extra volume, and this can be modest.
2 copies minimum !!!: An important point to note is that if you are serious about retaining your images then you should have 2 copies in existence as soon as possible after taking the photo and then at all times from then on. ANY storage system that has only one copy will fail you sooner or later - be it hard drive, DVD ROM, SSD, flash memory stick or card, printed image or other.
SSD versus Flash: While SSD's, as recommended by some, are indeed a possible solution, you can do very well using high capacity flash memory devices - either as "memory sticks" or as plug in memory cards. SSD's and flash use the same underlying technology - in the case of the invividual cards you become a greater part of the control system than with the SSD. This can be less convenient but has some major advantages. A micro-SD or T-flash card weighs about 0.25 gram ! with dimensions of 15 x 11 x 1 mm. 100 of these weigh 25 grams (just under 1 ounce) and fit in about 16vv / 1 cubic inch "close packed". They would hold 1.6 GB of data using 16 GB cards. Larger versions are available but generally not recommended. Pricing should be competitive with an SSD. Speed need not be high. Using eg Class 4 cards would be fine for this application - reliability and a known good brand would rate ahead of speed. If two copies are held in each case, the advantage over an SSD is that in the event of data loss from card failure, only a small portion of one of the two copies is lost and may be replicated if detected. With an SSD, if failure is catastrophic at the card level, which is always possible, then the whole image needs to be rewritten and you are down to one only copy until a new version is made.
Hard drives: All modern hard-drives "park" their heads when not in use. THey have published shock survival specs and a modern drive is surprisingly good when the heads are parked. In most modern laptops, tablets etc head parking is initiated if freefall is detected - push you netbook off the table and the heads should be happily parked long before floor impact. Deceleration in g's is measured simply by (drop distance / stopping distance). Drop a drive one metre onto concrete and have about 1mm of travel available in case and mounting flexure and you get a 1000g stop. Ouch!. BUT if you can make the stopping distance 10mm then you get a 100g drop. If you can make that 100mm it's 10g. 100mm is a bit hard usually BUT 10mm is very easy. Wrap a drive in a towel and then in a layer of dense foam or small-bubble bubble-wrap and it's well on the way to being drop if not bullet proof. Very firm packing may not flex enough. Very soft packing may flax and "bottom" and still impart shock. A graded material (eg towel plus bubble wrap) probably does fairly well. Something like a few layers of urethane closed cell material as used in wetsuits and a little thought could result in a superb level of protection. Many modern backpacks have a padded pocket closest to the wearers back and a modern laptop stored in that pocket will probably live through a super high speed incident even if the wearer does not :-(.
What I do: I have travelled to and in Asia 15 times in the last 5 years, including reasonably extensive travel within China. I take many many photos.
Multiple copies: I said above that two+ copies of all photos are needed. I usually carry a netbook with me. During a day when time allows I start the netbook and make an incremental download to netbook hard disk. I do this with XXCOPY (olde steam driven DOS program) but use what suits. The master copy stays on the camera card but a flag is set to show it has been copied. At the end of the day I copy (not move) from netbook to external hard drive. I now have 3 copies. (Camera card, netbook, external drive). On some occasions I tape (literally) a harddrive to the rear of the netbook screen to allow backup to external drive along the way.
I typically carry about 50 GB of camera memory cards. Depending on length of travel and activities undertaken these can last for some days without deleting, providing an ongoing 3rd (or 4th) copy to the ones on netbook and external drive(s).
Worst case ever, somewhere between or at Shanghai and Brunei I lost 2 hard drives, and most of my memory cards, presumably to an opportunistic stranger. On that occasion I (stupidly) did NOT have a "knee-drive". I lost no photos. My final netbook based copy saved the day. The knee-drive gets more strictly used since then.