Light scattering imposes a major obstacle for imaging objects seated deeply in turbid media, such as biological tissues and foggy air. Diffuse optical tomography (DOT) tackles scattering by volumetrically recovering the optical absorbance and has shown significance in medical imaging, remote sensing and autonomous driving. A conventional DOT reconstruction paradigm necessitates discretizing the object volume into voxels at a pre-determined resolution for modelling diffuse light propagation and the resulting spatial resolution of the reconstruction is generally limited. We propose NeuDOT, a novel DOT scheme based on neural fields (NF) to continuously encode the optical absorbance within the volume and subsequently bridge the gap between model accuracy and high resolution. Comprehensive experiments demonstrate that NeuDOT achieves submillimetre lateral resolution and resolves complex 3D objects at 14 mm-depth, outperforming the state-of-the-art methods. NeuDOT is a non-invasive, high-resolution and computationally efficient tomographic method, and unlocks further applications of NF involving light scattering.