It might just be theoretically possible to convert the DNA of a human embryo at the single celled stage with a sufficient number of retroviral steps. It would be completely impractical to do it though. We're only just experimenting with retroviruses to insert small pieces of DNA, and we don't have an awful lot of control over the process. Since you'd basically be ending up with an animal embryo at the end of the process it would be pointless anyway.
It wouldn't work with an adult. Your DNA is important, but there's only so much it can do. Even if you somehow edited the DNA of every cell in your body to that of, say, a cat you'd still look mostly human. You might get fur at a pinch, but it wouldn't give you a tail, claws, etc or any change in the skeleton and hence overall body shape. Your body is already grown, from a human template. Plus the incompatible biochemistry between the cat DNA and the human proteins making up your cells would kill you.
I'd file this idea as "not conclusively proven impossible". It would take huge advances in genetic engineering, a total understanding of how human and animal biology works, and the ability to extend that to make large scale changes to an adult body possible. It would almost certainly be more difficult than e.g. figuring out how to cure all diseases.