In this paper we consider the mutual exclusion problem on a multiple access channel. Mutual exclusion is one of the fundamental problems in distributed computing. In the classic version of this problem, n processes perform a concurrent program which occasionally triggers some of them to use shared resources, such as memory, communication channel, device, etc. The goal is to design a distributed algorithm to control entries and exits to/from the shared resource in such a way that in any time there is at most one process accessing it. We consider both the classic and a slightly weaker version of mutual exclusion, called ε-mutual-exclusion, where for each period of a process staying in the critical section the probability that there is some other process in the critical section is at most ε. We show that there are channel settings, where the classic mutual exclusion is not feasible even for randomized algorithms, while ε-mutual-exclusion is. In more relaxed channel settings, we prove an exponential gap between the makespan complexity of the classic mutual exclusion problem and its weaker ε-exclusion version. We also show how to guarantee fairness of mutual exclusion algorithms, i.e., that each process that wants to enter the critical section will eventually succeed.