It's certainly not necessary to have the CompTIA (or any other) qualification prior to starting on CISSP.
Many people come to broad ranging security roles from either technical security (i.e. firewall or network admin) or from general IT. Others come in from more general employment in high-security environments (e.g. military and some government) or from specialist roles such as risk, audit or forensics, where they gain a degree of security experience.
Others come straight from college?
Personally, I was a systems engineer in a (very) high-security environment, and my last role project managing a secure communications system. I then transitioned in to security testing and then in to more general security work.
It is possible to take the test and pass with good preparation. However, if you are just getting into security it may be a good idea to do the Security+ or SSCP first. The requirements for the SSCP are must less stringent than the CISSP and once you pass it you can join ISC2. Either way Good Luck!!!
I don't have a degree in any security field.
Neither did I. I have a degree in electrical and electronic engineering. No security content whatsoever.
Taking the CISSP course straight without any knowledge in security, is it a good start?
For people without any academic background in computer security, I've always recommended Dieter Gollman's "Computer Security" book. It covers all the dull stuff (formal security models etc) that you are highly unlikely to have to think about after the exam. However, BoK revisions may have made this less important. Security Engineering (Ross Anderson) is also a solid background.