Rule-based languages lie at the core of several areas of central importance to databases and artificial intelligence, such as data exchange, deductive databases, and knowledge representation and reasoning, to name a few. An algorithmic task that involves rules and is crucial for the above applications is query answering. In this work, we perform an in-depth complexity analysis of query answering under the main guarded-based classes of disjunctive tuple-generating dependencies (TGDs), focussing on (unions of) conjunctive queries ((U)CQs). We show that the problem under investigation is very hard, namely 2EXPTIME-complete, even for fixed sets of dependencies of a very restricted form. This is a surprising lower bound that demonstrates the enormous impact of disjunction on query answering under guarded-based TGDs, and also reveals the source of complexity for expressive computational logics such as the guarded fragment of first-order logic. We then proceed to investigate whether prominent subclasses of (U)CQs, i.e., queries of bounded treewidth and hypertree-width, and acyclic queries, have a positive impact on the complexity of the problem under consideration. Interestingly, we show that queries of bounded treewidth and of bounded hypertree-width do not reduce the complexity of our problem, even if we focus on predicates of bounded arity, or on fixed sets of disjunctive TGDs. Regarding acyclic queries, although the problem remains 2EXPTIME-complete in general, in some relevant settings the complexity reduces to EXPTIME-complete; in fact, this requires to bound the arity of the predicates, and for some expressive guarded-based formalisms, to fix the set of dependencies. Finally, with the aim of identifying tractable cases, we focus our attention on atomic queries. We show that atomic queries do not make the query answering problem easier under classes of guarded-based disjunctive TGDs that allow more than one atom to occur in the body of the dependencies. However, the complexity significantly decreases in the case of dependencies that can have only one atom in the body, and in fact we obtain the first (and, to the best of our knowledge, the only known) tractability results for query answering under disjunctive TGDs. In particular, we obtain a PTIME-completeness if we focus on predicates of bounded arity, and AC0-membership when the set of dependencies and the query are fixed. It is worth noting that our techniques and results on query answering under guarded disjunctive TGDs can be used as a generic tool for establishing complexity results for query answering under central description logics.