A fundamental challenge in Continual Learning (CL) is catastrophic forgetting, where adapting to new tasks degrades the performance on previous ones. While the field has evolved with diverse methods, this rapid surge in diverse methodologies has culminated in a fragmented research landscape. The lack of a unified framework, including inconsistent implementations, conflicting dependencies, and varying evaluation protocols, makes fair comparison and reproducible research increasingly difficult. To address this challenge, we propose LibContinual, a comprehensive and reproducible library designed to serve as a foundational platform for realistic CL. Built upon a high-cohesion, low-coupling modular architecture, LibContinual integrates 19 representative algorithms across five major methodological categories, providing a standardized execution environment. Meanwhile, leveraging this unified framework, we systematically identify and investigate three implicit assumptions prevalent in mainstream evaluation: (1) offline data accessibility, (2) unregulated memory resources, and (3) intra-task semantic homogeneity. We argue that these assumptions often overestimate the real-world applicability of CL methods. Through our comprehensive analysis using strict online CL settings, a novel unified memory budget protocol, and a proposed category-randomized setting, we reveal significant performance drops in many representative CL methods when subjected to these real-world constraints. Our study underscores the necessity of resource-aware and semantically robust CL strategies, and offers LibContinual as a foundational toolkit for future research in realistic continual learning. The source code is available from \href{https://github.com/RL-VIG/LibContinual}{https://github.com/RL-VIG/LibContinual}.