JDH has given a deep, interesting answer -- but it's deep and interesting in part because it relates to ZFC, which is a deep and interesting theory. Formal mathematical theories don't have to be deep and interesting, and it is possible to answer this question using very simple and straightforward examples.
Here is an example of a mathematical theory:
There are three well-formed formulas in this theory: x, y, and z. We have an axiom that says that x is true, and another axiom that says y is false. That's it. This theory has no grammatical rules for producing more complex formulas from simpler ones, no other machinery. In this theory, z cannot be assigned a truth value.
A less trivial example is the following. Take Tarski's axioms and delete the axiom of Euclid. This is a formal system that represents the same ideas as Euclid's original formulation of plane geometry, but without the parallel postulate. In this system, we have various statements that cannot be assigned truth values. One such statement is the axiom of Euclid (i.e., basically the parallel postulate). Another would be the Pythagorean theorem.
The self-referential thing is an interpretation of a particular strategy used by Godel for constructing undecidable statements in theories that can describe a certain amount of arithmetic. Note the three parts: (1) an interpretation, (2) a particular strategy, and (3) only for theories that can describe a certain amount of arithmetic.
The examples I've given above don't require Godel's strategy. Furthermore, Godel's strategy doesn't even work for Tarski's system, because Tarski's system can't describe the necessary amount of arithmetic.
Even in examples that do use Godel's strategy, the self-referentialism is only an interpretation. The undecidable statements don't literally refer to themselves -- they can just be interpreted that way.