This paper is the first to present a novel, non-contact method that utilizes orthogonal frequency division multiplexing (OFDM) signals (of frequency 5.23 GHz, emitted by a software defined radio) to radio-expose the pulmonary patients in order to differentiate between five prevalent respiratory diseases, i.e., Asthma, Chronic obstructive pulmonary disease (COPD), Interstitial lung disease (ILD), Pneumonia (PN), and Tuberculosis (TB). The fact that each pulmonary disease leads to a distinct breathing pattern, and thus modulates the OFDM signal in a different way, motivates us to acquire OFDM-Breathe dataset, first of its kind. It consists of 13,920 seconds of raw RF data (at 64 distinct OFDM frequencies) that we have acquired from a total of 116 subjects in a hospital setting (25 healthy control subjects, and 91 pulmonary patients). Among the 91 patients, 25 have Asthma, 25 have COPD, 25 have TB, 5 have ILD, and 11 have PN. We implement a number of machine and deep learning models in order to do lung disease classification using OFDM-Breathe dataset. The vanilla convolutional neural network outperforms all the models with an accuracy of 97%, and stands out in terms of precision, recall, and F1-score. The ablation study reveals that it is sufficient to radio-observe the human chest on seven different microwave frequencies only, in order to make a reliable diagnosis (with 96% accuracy) of the underlying lung disease. This corresponds to a sensing overhead that is merely 10.93% of the allocated bandwidth. This points to the feasibility of 6G integrated sensing and communication (ISAC) systems of future where 89.07% of bandwidth still remains available for information exchange amidst on-demand health sensing. Through 6G ISAC, this work provides a tool for mass screening for respiratory diseases (e.g., COVID-19) at public places.