Multi-access Edge Computing (MEC) has recently emerged as a potential technology to serve the needs of mobile devices (MDs) in 5G and 6G cellular networks. By offloading tasks to high-performance servers installed at the edge of the wireless networks, resource-limited MDs can cope with the proliferation of the recent computationally-intensive applications. In this paper, we study the computation offloading problem in a massive multiple-input multiple-output (MIMO)-based MEC system where the base stations are equipped with a large number of antennas. Our objective is to minimize the power consumption and offloading delay at the MDs under the stochastic system environment. To this end, we formulate the problem as a Markov Decision Process (MDP) and propose two Deep Reinforcement Learning (DRL) strategies to learn the optimal offloading policy without any prior knowledge of the environment dynamics. First, a Deep Q-Network (DQN) strategy to solve the curse of the state space explosion is analyzed. Then, a more general Proximal Policy Optimization (PPO) strategy to solve the problem of discrete action space is introduced. Simulation results show that the proposed DRL-based strategies outperform the baseline and state-of-the-art algorithms. Moreover, our PPO algorithm exhibits stable performance and efficient offloading results compared to the benchmark DQN strategy.