Intensity diffraction tomography (IDT) refers to a class of optical microscopy techniques for imaging the 3D refractive index (RI) distribution of a sample from a set of 2D intensity-only measurements. The reconstruction of artifact-free RI maps is a fundamental challenge in IDT due to the loss of phase information and the missing cone problem. Neural fields (NF) has recently emerged as a new deep learning (DL) paradigm for learning continuous representations of complex 3D scenes without external training datasets. We present DeCAF as the first NF-based IDT method that can learn a high-quality continuous representation of a RI volume directly from its intensity-only and limited-angle measurements. We show on three different IDT modalities and multiple biological samples that DeCAF can generate high-contrast and artifact-free RI maps.