Machine learning is a powerful approach for fitting microstructural models to diffusion MRI data. Early machine learning microstructure imaging implementations trained regressors to estimate model parameters in a supervised way, using synthetic training data with known ground truth. However, a drawback of this approach is that the choice of training data impacts fitted parameter values. Self-supervised learning is emerging as an attractive alternative to supervised learning in this context. Thus far, both supervised and self-supervised learning have typically been applied to isotropic models, such as intravoxel incoherent motion (IVIM), as opposed to models where the directionality of anisotropic structures is also estimated. In this paper, we demonstrate self-supervised machine learning model fitting for a directional microstructural model. In particular, we fit a combined T1-ball-stick model to the multidimensional diffusion (MUDI) challenge diffusion-relaxation dataset. Our self-supervised approach shows clear improvements in parameter estimation and computational time, for both simulated and in-vivo brain data, compared to standard non-linear least squares fitting. Code for the artificial neural net constructed for this study is available for public use from the following GitHub repository: https://github.com/jplte/deep-T1-ball-stick.