In the smart mariculture, batch testing of breeding traits is a key issue in the breeding of improved fish varieties. The body length (BL), body width (BW) and body area (BA) features of fish are important indicators. They are of great significance in breeding, feeding and classification. To accurately and intelligently obtain the morphological characteristic sizes of fish in actual scenes, data augmentation is first used to greatly expand the published fish dataset, thereby ensuring the robustness of the training model. Then, an improved U-net segmentation and measurement algorithm is proposed, which uses a dilated convolution with a dilation rate 2 and a convolution to partially replace the convolution in the original U-net. This operation can enlarge the partial convolution receptive field and achieve more accurate segmentation for large targets in the scene. Finally, a line fitting method based on the least squares method is proposed, which is combined with the body shape features of fish and can accurately measure the BL and BW of inclined fish. Experimental results show that the Mean Intersection over Union (mIoU) is 97.6% and the average relative error of the area is 0.69%. Compared with the unimproved U-net, the average relative error of the area is reduced to about half. Moreover, with the improved U-net and the line fitting method, the average relative error of BL and the average relative error of BW of inclined fish decrease to 0.37% and 0.61%, respectively.