Is it possible for a physical object to be four-dimensional? This depends a bit on what you mean by a "physical object". But it seems plausible to say that the could be a four-dimensional world with four-dimensional objects in it, and why should we not in talking about such a possible world call such objects physical objects? I would address your question in the same way. There could be a one-dimensional world, and there could be objects in it. Again, it seems plausible to say of these one-dimensional objects that they are the physical objects of that one-dimensional world. But then again, if someone where to feel very strongly that the expression "physical object" should be reserved for three-dimensional objects, I for one would not want to get into a long argument over this point.
Now another question you may have in mind is whether there can be one-dimensional objects in our three-dimensional space. We can certainly describe such objects geometrically, so they are possible in this sense (and, again, the question whether we should call such an object a physical object seems to me to be of little interest). Whether the existence of such objects is consistent with the natural laws of our universe and, if so, whether we could detect them, are different questions which, I think, you are not asking.