Of course self-replicating probes are possible. If we built the first spaceship, we can build more. Put some of us in the spaceship. When we get to our first destination, we do the same things we did to make the first one, again.
Or we make other self-replicating machines instead of us, if that turns out to be more practical. Again, we know that self-replicating machines are possible, because we’re examples of such.
And sure, we don’t have the technology to do that right now. But if we have the technology in a thousand years from now, or ten thousand, that’s good enough.