Genetically the chromosomes don't match, and it's considered off limits to even do research into it. If we attempted to mate dog sperm to a human egg, the Zygote would die off within the second division.
However, we do have the knowledge to implant fertilized embryos' from one species into another species womb and carry the pregnancy to term. Cloning is done this way sometimes.
Theoretically we could totally mess up your body chemistry, suppress your body's natural selection protections, and implant fertilized dog eggs into your uterus and have you carry them to term. Now no one is going to do this, and you might well die from the process, but it's out there.