You could certainly rack mount your own case if you want. First though, it must be small enough to fit. Second, you ahve to find good rack ears. Third, you have to find a way to SECURELY mount them. Fourth, you have to make sure that they are perfectly aligned to keep things form vibrating too much. Fifth, you will probably wnat to figure out how to at least get your cd/dvd drives hrizontal again. In the end, buying the right case seems like a MUCH better option, and probably still cheaper and more functional as well.
As far as hard drives go, they certainly do not have to be horizontal. It is helpful to always keep them level though (either horizontal or vertical).
Keep in mind, that rack stability is critical in a rack mounted PC. If the rack itself vibrates for any reason (which is common) you will be putting extra stress on all of your components, especially your hard drives. Also, when a rack vibrates like that, it will often times make a computer louder than if it were not racked.....