We propose and study a realistic Continual Learning (CL) setting where learning algorithms are granted a restricted computational budget per time step while training. We apply this setting to large-scale semi-supervised Continual Learning scenarios with sparse label rate. Previous proficient CL methods perform very poorly in this challenging setting. Overfitting to the sparse labeled data and insufficient computational budget are the two main culprits for such a poor performance. Our new setting encourages learning methods to effectively and efficiently utilize the unlabeled data during training. To that end, we propose a simple but highly effective baseline, DietCL, which utilizes both unlabeled and labeled data jointly. DietCL meticulously allocates computational budget for both types of data. We validate our baseline, at scale, on several datasets, e.g., CLOC, ImageNet10K, and CGLM, under constraint budget setup. DietCL outperforms, by a large margin, all existing supervised CL algorithms as well as more recent continual semi-supervised methods. Our extensive analysis and ablations demonstrate that DietCL is stable under a full spectrum of label sparsity, computational budget and various other ablations.