I would like to put Arch Linux on my 500 GB HDD. How would I do this, and is it possible to configure the install so that I only have to boot from the external disk on the target computer's BIOS? I don't want to have to put GRUB or anything. I'd like the freedom of a live USB but with the power of a fully installed system.
You would need to install grub on the external HDD as far as i know, because it will still need a boot loader. Connect it up press either F8 or F12 it's BIOS dependent and then you should be able to boot from the External HDD but only if the system allows booting from USB
Some BIOS software will give you the option of a Boot Menu you can enter which lets you choose a device to boot. Look for the prompt on boot, I typically see the key to use the menu being set to F8.