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5G and beyond cellular networks (NextG) will support the continuous execution of resource-expensive edge-assisted deep learning (DL) tasks. To this end, Radio Access Network (RAN) resources will need to be carefully "sliced" to satisfy heterogeneous application requirements while minimizing RAN usage. Existing slicing frameworks treat each DL task as equal and inflexibly define the resources to assign to each task, which leads to sub-optimal performance. In this paper, we propose SEM-O-RAN, the first semantic and flexible slicing framework for NextG Open RANs. Our key intuition is that different DL classifiers can tolerate different levels of image compression, due to the semantic nature of the target classes. Therefore, compression can be semantically applied so that the networking load can be minimized. Moreover, flexibility allows SEM-O-RAN to consider multiple edge allocations leading to the same task-related performance, which significantly improves system-wide performance as more tasks can be allocated. First, we mathematically formulate the Semantic Flexible Edge Slicing Problem (SF-ESP), demonstrate that it is NP-hard, and provide an approximation algorithm to solve it efficiently. Then, we evaluate the performance of SEM-O-RAN through extensive numerical analysis with state-of-the-art multi-object detection (YOLOX) and image segmentation (BiSeNet V2), as well as real-world experiments on the Colosseum testbed. Our results show that SEM-O-RAN improves the number of allocated tasks by up to 169% with respect to the state of the art.