Not a scholar of the period, but I was under the impression that Greece's democratic system meant that only male land-owners could vote and that the leading philosophers of the day considered slavery the natural order of things, women marginally (if at all) better than slaves, and boys suitable for divine pleasure. It doesn't seem particularly democratic for them?
I do think it's reasonable to see in ancient Greece the seeds for democracy, but modern republican and democratic forms are pretty far removed from those days; and I don't suppose most of us would be very pleased to live under such a system.