OpenBSD has to be a BSD Unix and you couldn't duplicate it with Linux
OpenBSD has a well deserved reputation for putting security and a
clean system (for code, documentation, and so on) first, and
everything else second. OpenBSD is of course based on BSD (it's
right there in the name) and descends from
NetBSD (you can read the history here). But one of the questions
you could ask about it is whether it had to be that way, and in
particular if you could build something like OpenBSD on top of
Linux. I believe that the answer is no.
Linux and the *BSDs have a significantly different model of what they are. BSDs have a 'base system' that provides an integrated and fully operational core Unix, covering the kernel, C library and compiler, and the normal Unix user level programs, all maintained and distributed by the particular BSD. Linux is not a single unit this way, and instead all of the component parts are maintained separately and assembled in various ways by various Linux distributions. Both approaches have their advantages, but one big one for the BSD approach is that it enables global changes.
Making global changes is an important part of what makes OpenBSD's approach to improving security, code maintenance, and so on work. Because it directly maintains everything as a unit, OpenBSD is in a position to introduce new C library or kernel APIs (or change them) and then immediately update all sorts of things in user level programs to use the new API. This takes a certain amount of work, of course, but it's possible to do it at all. And because OpenBSD can do this sort of ambitious global change, it does.
This goes further than just the ability to make global changes, because in theory you can patch in global changes on top of a bunch of separate upstream projects. Because OpenBSD is in control of its entire base system, it's not forced to try to reconcile different development priorities or integrate clashing changes. OpenBSD can decide (and has) that only certain sorts of changes will be accepted into its system at all, no matter what people want. If there are features or entire programs that don't fit into what OpenBSD will accept, they just lose out.
(I suspect that this decision on priorities gives OpenBSD has more leverage to push other people in directions that it wants, because the OpenBSD developers are clearly willing to remove support for something if they feel strongly enough about it. For example, I suspect that their new system call origin verification is going to eventually force Go to make system calls only through OpenBSD's C library, contrary to what Go prefers to do.)