Now that Docker does not rely on LXC anymore, would it be possible to provide native OS X support?

Or would it be impossible as it will need such BSD features as jails, etc. which are missing in OS X?

It is indeed possible for Docker to be ported to other exec drivers. The freebsd project recently release a version using jails + freebsd’s linux executable support:

I don’t know Darwin internals too well, but it is true that they don’t include Jails. If they had any sort of container or sandbox support, it is indeed theoretically possible to get a docker implementation working on top of that.