You can connect either an external HDD or USB thumb drive up to 16GB. The hard drive is not essential and can easily be replaced with a USB thumb drive. The HDD serves only a few purposes:
1.) Retail games (they sell full retail Xbox 360 games on Xbox Live). These are usually more expensive than the disc based copies and you can't resell them so I don't recommend it personally. Typical sizes are about 5GB-7GB
2.) Game Demos. Usually deleted right after playing. Ranges in size about 700MB-2GB.
3.) Game add-ons. These include extra maps for multiplayer games, additional campaign levels or missions, new songs for a music game or new in-game characters. These could range from a few megabytes to a couple of gigabytes depending upon the amount of content you are buying. Extra multiplayer levels or missions could use a lot of space.
4.) Xbox Live Arcade Games or Indie Games. These range from usually really small 40MB to the occasional big game 300MB. None of these games require a HDD, but they could add up over time. You will be fine with a USB thumb drive for these games though. And you can always delete and redownload later if you need to clear space.
5.) Installing disc based retail games to the HDD. This was much more useful with the old Xbox 360 because the DVD drive was so loud when it would spin. The new Xbox 360 is so quiet that there is very little advantage to installing games. Load times are only reduced by a few seconds by installing games to the HDD.
6.) Game videos. I expect this will eventually change, but for some reason unlike movies, movie trailers, tv shows, and music videos Microsoft still requires you to download game trailers to your hard drive. Microsoft clearly has the ability to stream 1080p video so there is no reason for this any longer. For now though you need storage space to watch game trailers.
You don't need a hard drive for Netflix, Zune video, Zune music, Last.FM, Facebook, Twitter, ESPN, or Hulu. All of this content streams off the internet and is not stored locally on the system.
Basically the only thing you NEED a hard drive for is if you expect to download a lot of extra content (DLC) like multiplayer map packs for Call of Duty or Halo or if you expect to buy a ton of songs for music games.