Is it possible to build a computer that could run any program from any OS?

Yes, Alan Turing came up with a way of thinking physically about logic. You can take any well-defined, step-by-step logical process and create a machine to do it (we call these Turing machines today). Some can be simple processes, like addition, but they aren't limited to being simple.

He also proved that it was possible to make a Universal Turing machine which could do the work of any other Turing machine. When you can take the input for another machine, all you theoretically have to do is use the transition rules created for the other machine and you have a machine that can read instructions. You have a programmable computer.

So this is not a new idea at all, but as we moved from one technology (core memory to vacuum tubes to RAM; punch cards to floppies to hard disks; kilohertz to megahertz to gigahertz) we don't always replace the old tech with new stuff that is sufficiently fast enough to completely simulate the old stuff in real time. We can probably emulate 1970's tech pretty easily today, but maybe aren't yet up to 1990's tech without overlap in the design or taking shortcuts in the logic.

Sometimes we can emulate old tech perfectly but we may not want to. We can create a machine that computes with the same programs of WWII era mainframes, but do we really want a room that sounds like "a million knitting needles" constantly clicking and ticking to compute the answers? Maybe it's nostalgic; maybe it's just maddening. Some old tech is nostalgic (like the cheats 70's era computer makers used to get around limits of NTSC and PAL televisions) but it still may not be worth the effort to emulate the output perfectly. Different programmers will establish will come up with different answers to what is worth the effort. But a universal (even self-reproducing) Universal Turing machine is theoretically possible.

tl;dr: Because different people have different priorities about what is worth simulating, we will always have different emulators at different stages of development.