Accurate segmentation of the liver parenchyma, portal veins, hepatic veins, and lesions from MRI is important for hepatic disease monitoring and treatment. Multi-phase contrast enhanced imaging is superior in distinguishing hepatic structures compared to single-phase approaches, but automated approaches for detailed segmentation of hepatic structures are lacking. This study evaluates deep learning architectures for segmenting liver structures from multi-phase Gd-EOB-DTPA-enhanced T1-weighted VIBE MRI scans. We utilized 458 T1-weighted VIBE scans of pathological livers, with 78 manually labeled for liver parenchyma, hepatic and portal veins, aorta, lesions, and ascites. An additional dataset of 47 labeled subjects was used for cross-scanner evaluation. Three models were evaluated using nested cross-validation: the conventional nnU-Net, the ResEnc nnU-Net, and the Swin UNETR. The late arterial phase was identified as the optimal fixed phase for co-registration. Both nnU-Net variants outperformed Swin UNETR across most tasks. The conventional nnU-Net achieved the highest segmentation performance for liver parenchyma (DSC: 0.97; 95% CI 0.97, 0.98), portal vein (DSC: 0.83; 95% CI 0.80, 0.87), and hepatic vein (DSC: 0.78; 95% CI 0.77, 0.80). Lesion and ascites segmentation proved challenging for all models, with the conventional nnU-Net performing best. This study demonstrates the effectiveness of deep learning, particularly nnU-Net variants, for detailed liver structure segmentation from multi-phase MRI. The developed models and preprocessing pipeline offer potential for improved liver disease assessment and surgical planning in clinical practice.