The ever-increasing demand for ubiquitous, continuous, and high-quality services poses a great challenge to the traditional terrestrial network. To mitigate this problem, the mobile-edge-computing-enhanced low earth orbit (LEO) satellite network, which provides both communication connectivity and on-board processing services, has emerged as an effective method. The main issue in LEO satellites includes finding the optimal locations to host network functions (NFs) and then making offloading decisions. In this article, we jointly consider the problem of service chain caching and computation offloading to minimize the overall cost, which consists of task latency and energy consumption. In particular, the collaboration among satellites, the network resource limitations, and the specific operation order of NFs in service chains are taken into account. Then, the problem is formulated and linearized as an integer linear programming model. Moreover, to accelerate the solution, we provide a greedy algorithm with cubic time complexity. Numerical investigations demonstrate the effectiveness of the proposed scheme, which can reduce the overall cost by around 20% compared to the nominal case where NFs are served in data centers.