As someone who used to work in electronics manufacturing, I can confirm that the concerns mentioned by the others would be extremely difficult and costly to overcome. The driver board would be far more complex, and the multiple heads would have to be 100% accurate and perfectly synced to avoid errors. This is nearly impossible without the highest grade components you can get (~0% manufacturing tolerance). I'm sure it could be done, but the hard drive would cost more than the computer, and SSD prices per GB have already dropped around 75% from where they were a few years ago. It would also still be slower than an SSD since it takes time for the mechanical components to reach the destination of the stored data, where an SSD has almost immediate access to every stored bit.
So, yes, someone could probably make a hard drive the way that you've suggested, but it would unfortunately be impractical. It's also unlikely that it would fit well in a laptop or tablet.