That would mean something a bit different: "She does not want the shirt."

The version with the definite article (the/το) implies that you have already talked about a specific shirt before, and the listener knows which one you mean. Then in this sentence you say that she does not want this particular shirt which is known to the listener.

But with the indefinite article (a/ένα), you are not speaking about any specific shirt. That sentence says that she does not any shirt at all.