Authentication of wireless nodes, as in fifth-generation (5G) and Internet of Things (IoT) networks, is an increasingly pressing issue, in order to limit the required computational effort and the necessary overhead. A simplification of the authentication process may therefore be of interest to achieve the satisfaction of stringent performance requirements, such as those envisaged for sixth-generation (6G) networks. This paper provides a study on the feasibility of physical layer authentication (PLA) in a real indoor environment, as an alternative solution to the traditional authentication schemes. To ensure the reliability of the proposed approach a simulated scenario is firstly tested. Subsequently, real-world data are collected through a laboratory setup using a Vectorial Signal Transceiver (VST) and two Universal Software Radio Peripherals (USRPs) to emulate the behavior of the receiver, the legitimate transmitter, and the potential adversary. A machine learning (ML) algorithm is then exploited to act as authenticator. This means that channel fingerprint is extracted from signals to create a dataset used to train a sparse autoencoder. To emulate areal authentication scenario, the autoencoder is trained only on the class of the legitimate user. Once anew message arrives, the autoencoder task is to discern authentic signals from those forged by the adversary. It is shown that a geometric mean of accuracy of more than 90%, with corresponding low levels of false alarm and missed detection, is achievable irrespective of the nodes location, underlining the robustness and versatility of the proposed ML-based PLA approach.