Coded Aperture γ-cameras have been extensively used in applications ranging from astrophysics to nuclear medicine for imaging radioactive source distributions. These devices allow the identification of the direction of γ-emitters by analyzing the shadow patterns projected onto pixelated detectors. In this work, we propose a novel approach based on Convolutional Neural Networks (CNNs) for accurately localizing radioactive sources in 3D using one coded aperture gamma camera. The CNN is trained using simulated shadowgrams generated by a custom simulation tool, with sources placed at various positions within the near-field, ranging from 20 cm to 120 cm from the detector. Unlike previous methods that focused on estimating angular coordinates, our model also accurately estimates the distance of the source, achieving a distance estimation accuracy of 10 mm, in addition to determining the polar and azimuthal angles. This capability is particularly relevant for medical imaging applications, where precise localization of radioactive sources is crucial. The results of our study demonstrate the potential of CNN-based algorithms in improving the accuracy and reliability of single-detector systems in near-field radioactive source localization.