The BeOS is not Unix- or Mach-based (we're a little sensitive about being compared with 1970s technology). It is a completely new operating system, written from the ground up.
While it's a new design, it does owe much to UNIX/Mach in some of the design decisions we made. If you're interested in some of the deeper details, check out the XINU (that's "Unix" spelled backwards, which cries out for a joke here...) book listed in The Be Bookstore:
The designers of the BeOS were deeply inspired by the design of the XINU operating system.