The clinical value of quantitative MRI hinges on its measurement repeatability. Deep learning methods to reconstruct undersampled quantitative MRI can accelerate reconstruction but do not aim to promote quantitative repeatability. This study proposes a repeatability-encouraging self-supervised learning (SSL) reconstruction method for quantitative MRI. The proposed SSL reconstruction network minimized cross-data-consistency between two equally sized, mutually exclusive temporal subsets of k-t-space data, encouraging repeatability by enabling each subset's reconstruction to predict the other's k-t-space data. The method was evaluated on cardiac MR Multitasking T1 mapping data and compared with supervised learning methods trained on full 60-s inputs (Sup60) and on split 30-s inputs (Sup30/30). Reconstruction quality was evaluated on full 60-s inputs, comparing results to iterative wavelet-regularized references using Bland-Altman limits of agreement (LOAs). Repeatability was evaluated by splitting the 60-s data into two 30-s inputs, evaluating T1 differences between reconstructions from the two halves of the scan. On 60-s inputs, the proposed method produced comparable-quality images and T1 maps to the Sup60 method, with T1 values in general agreement with the wavelet reference (LOA Sup60 = ±75 ms, SSL = ±81 ms), whereas the Sup30/30 method generated blurrier results and showed poor T1 agreement (LOA Sup30/30 = ±132 ms). On back-to-back 30-s inputs, the proposed method had the best T1 repeatability (coefficient of variation SSL = 6.3%, Sup60 = 12.0%, Sup30/30 = 6.9%). Of the three deep learning methods, only the SSL method produced sharp and repeatable images. Without the need for labeled training data, the proposed SSL method demonstrated superior repeatability compared with supervised learning without sacrificing sharpness, and reduced reconstruction time versus iterative methods.