The remarkable complexity of modern applied problems often requires the use of probabilistic models where the likelihood is intractable -- in the sense that it cannot be numerically evaluated, not even up to a normalizing constant. The statistical literature provides an extensive array of methods designed to bypass this constraint. Still, inference in this context remains computationally challenging, particularly for high-dimensional models. We focus on the important class of Approximation Bayesian Computation (ABC) methods. Various state-of-the-art ABC techniques are combined to fit an intractable model that describes the epidemiological dynamics of multidrug-resistant tuberculosis. This study addresses a number of important biological questions in a principled manner, providing useful insights to this extraordinarily relevant research topic. We propose a functional regression adjustment ABC procedure that permits the estimation of infinite-dimensional parameters, which effectively launches ABC into the non-parametric framework. Two likelihood-free algorithms are also introduced. The first exploits the principles of ABC and the so-called coverage property to recalibrate an auxiliary approximate posterior estimator. This approach further strengthens the links between ABC and indirect inference, allowing a more comprehensive use of the auxiliary estimator. The second algorithm employs the ABC machinery to build approximate samplers for the intractable full conditional distributions. These samplers are then combined to form a likelihood-free approximate Gibbs sampler. The granular nature of our approach (that comes from breaking down the problem into small pieces) makes it suitable for highly-structured problems. We demonstrate this property by fitting an intractable and very high-dimensional state space model.