Is it possible construct a tub out of tile?

You would need to also build in a custom pan and coat that with tar before building the tub. But, I have seen entire bathrooms done up in tile, including the floor, walls, shower stall, and bath tub. It might not be as comfortable as a regular bathtub, though, since you have to sit on the grout lines. You would probably just be better off building the platform for the tub, tiling that, and then setting a pre-formed tub into it. I did that with a Kohler jacuzzi tub and it's a whole lot better than if I had it done with all tiles.