This paper studies the problem of federated training of a deep neural network (DNN) over a random access channel (RACH) such as in computer networks, wireless networks, and cellular systems. More precisely, a set of remote users participate in training a centralized DNN model using SGD under the coordination of a parameter server (PS). The local model updates are transmitted from the remote users to the PS over a RACH using a slotted ALOHA protocol. The PS collects the updates from the remote users, accumulates them, and sends central model updates to the users at regular time intervals. We refer to this setting as the RACH-FL setting. The RACH-FL setting crucially addresses the problem of jointly designing a (i) client selection and (ii) gradient compression strategy which addresses the communication constraints between the remote users and the PS when transmission occurs over a RACH. For the RACH-FL setting, we propose a policy, which we term the ''age-of-gradient'' (AoG) policy in which (i) gradient sparsification is performed using top-K sparsification, (ii) the error correction is performed using memory accumulation, and (iii) the slot transmission probability is obtained by comparing the current local memory magnitude minus the magnitude of the gradient update to a threshold. Intuitively, the AoG measure of ''freshness'' of the memory state is reminiscent of the concept of age-of-information (AoI) in the context of communication theory and provides a rather natural interpretation of this policy. Numerical simulations show the superior performance of the AoG policy as compared to other RACH-FL policies.