I am creating a world in which a civilization has achieved advanced spaceflight and has begone colonizing other planets in their solar system that is similar to ours, but with more planets. However, the twist is that this entire civilization is like a test subject. A sort of "Truman Show meets micro verse batteries" scenario where an artificial universe containing this one star system is created inside of a housing unit, and scientist of an extremely advanced race observe it to study how societies develop. Is it possible to have a solar system with three stars that only spin and do not move through space? If not, is a binary or single star system possible under these parameters?
If you have only one star, it can be virtually stationary. (It will get tugged around by the gravity of its planets, but this is such a small effect in comparison to the mass of a star that you can pretty much discount it.) If you have more than one, at least one will have to move more substantially.
There are two basic options in an orbit: two objects can coorbit (in which case they both fall inwards, but their angular momentum causes them to orbit around one another), or one object can be virtually stationary while another, much less massive object orbits around it. There is no situation in which all bodies can be stationary, assuming gravity works the way it does in our universe.
Given two stars, although both will be moving, this will not be apparent from the stars themselves: they will tend to tidally lock, so that if you were standing on the surface of one (assuming for the moment you could do that) you would see the other at a fixed position in the sky.
Three stars of wildly different masses could be arranged in a Trojan configuration where the smaller two orbit the larger one at the same distance and speed. Your stars would form a triangle, basically, that rotates around one point (the massive star). Conventionally, this requires something like a 10000:100:1 ratio of masses to be stable in the long term.
Given that the system is artificial, though, it doesn't necessarily need to be stable in the long run. If the experimenters started off their test subjects at a certain level of evolution (say beginning with a prehistoric but biologically modern humanity), they could fit this kind of experiment into a few tens or hundreds of thousands of years; some "unstable" orbits can persist that long, or longer, even though they will inevitably eventually break up.