I still don't think it is entirely clear what you are trying to do. It is very helpful if you use the words
host in your question as to help clarify what exactly you are talking about.
First - read up on how virtual storage works and what options are available to you in the online manual, chapter 5: http://www.virtualbox.org/manual/ch05.html
Second - clearly you want to install at least one operating system to your USB media, and really if you are installing the host and the guest to the USB media, it is a bit beyond the scope of this question. But once you have the host installed (wherever it is going to be) - it appears the guest will definitely be installed on the USB drive.
You had mentioned specifically you don't want to have a virtual disk file sitting on the drive, so I am guessing want the entire drive to function as a virtual drive. There is a special type of virtual disk called a "raw hard disk". You will still need a VMDK file sitting somewhere that tells VirtualBox which partition to use in RAW mode, so you may want to actually partition the flash drive. This may also be problematic depending on your host as the physical path to the drive/partition can change depending on how your storage is initialized (in particular, the order) - when you set this up, your flash drive might be /dev/sdc but next time you boot (or use it in a different computer) your drive might show up as /dev/sdb. Chapter 9 talks about the raw format in more detail: http://www.virtualbox.org/manual/ch09.html#rawdisk
It would be very helpful to help state your goals or what constraints you are working with to drive you to wanting this type of setup. Right now the best we can offer is some general advice, make assumptions about it and hopefully steer you down the right path. I considered flagging this question because as it is originally stated it just isn't answerable - but I do think it is salvageable with a little more input from you.