A Hard Wrap For Hard Drives?
The simple answer is, 'no'. The original model of the Xbox 360 (more on the 'S' model coming up later) comes included with no internal memory storage (unlike the original Xbox, which came with a hard drive built in as standard). To save anything for later use, including save games, music, downloaded trailers, levels, demos, player preferences, and community-created content from Xbox Live Marketplace, some sort of memory device MUST be present.
A Hard Drive On The Wallet?
If the optional detachable 'SATA' hard drive are a little out of budget for you (although, shop around, there are a number of sizes available at differing prices; 20 GB, 60 GB, 120 GB, 250 GB), maybe you could plump for the smaller but much more affordable Memory Units.
The memory units only come in one size which is 256Mb of memory. While this will be fine for the majority of your save games, to fully experience your Xbox's full potential, and perhaps download some demos, additional content for your games, or a few trailers, you will need the additional memory that only a hard drive can provide.
Benefits For Having A Hard Drive?
Another downside to not having a hard drive for your Xbox 360 is that you may not be able to play certain games. The way some games are programmed to run on your machine means they are in need of the extra power provided by a hard drive at all times. Some of these games include Halo 3, Football Manager and Final Fantasy.
You also will not be able to access the Xbox's online functionalities or 'Xbox Live' without a hard drive.
So Should I Get One Or Not?
So, in summary then, yes, you can save games if you don't have a hard drive but do have a memory unit. But, you will miss out on so much that the Xbox has to offer without a hard drive, including certain games and the online support. So, it is highly advised that you do fork out that little extra for one to enjoy your console as fully as possible.
One Last Thing...
However, in June 2010, Microsoft announced and released the 'S' model of the Xbox 360 console, a slimmer, revised design of the original console. It was similar in every way aside from appearance, and shared all the same specs as its original. However, all Xbox 360 'S' models come with hard drive storage built into the machine, and so if you are lucky enough to have one of these models, you don't need to worry about purchasing any kind of additional memory storage!