Scaling up model depth and size is now a common approach to raise accuracy in many deep learning (DL) applications, as evidenced by the widespread success of multi-billion or even trillion parameter models in natural language processing (NLP) research. Despite success in DL research and at major technology companies, broader practical adoption of such large models among domain scientists and businesses is still bottlenecked by GPU memory limits, high training costs, and low GPU availability, even on public clouds. Model selection needs further compound these resource challenges: users often need to compare dozens of models with different hyper-parameters or neural architectures to suit their specific task and dataset. In this paper, we present Hydra, a system designed to tackle such challenges by enabling out-of-the-box scaling for multi-large-model DL workloads on even commodity GPUs in a resource-efficient manner. Hydra is the first approach to holistically optimize the execution of multi-model workloads for large DL models. We do this by adapting prior "model-parallel" execution schemes to work with scalable parameter offloading across the memory hierarchy and further hybridizing this approach with task-parallel job scheduling techniques. Hydra decouples scalability of model parameters from parallelism of execution, thus enabling DL users to train even a 6-billion parameter model on a single commodity GPU. It also fully exploits the speedup potential of task parallelism in multi-GPU setups, yielding near-linear strong scaling and making rigorous model selection perhaps more practical for such models. We evaluate end-to-end performance by fine-tuning GPT-2 for language modeling. We find that Hydra offers between 50% and 100% higher training throughput than even the best settings of state-of-the-art industrial frameworks such as DeepSpeed and GPipe for multi-large-model training.