We introduce a novel formulation for continuous space-time video super-resolution. Instead of decoupling the representation of a video sequence into separate spatial and temporal components and relying on brittle, explicit frame warping for motion compensation, we encode video as a continuous, spatio-temporally coherent 3D Video Fourier Field (VFF). That representation offers three key advantages: (1) it enables cheap, flexible sampling at arbitrary locations in space and time; (2) it is able to simultaneously capture fine spatial detail and smooth temporal dynamics; and (3) it offers the possibility to include an analytical, Gaussian point spread function in the sampling to ensure aliasing-free reconstruction at arbitrary scale. The coefficients of the proposed, Fourier-like sinusoidal basis are predicted with a neural encoder with a large spatio-temporal receptive field, conditioned on the low-resolution input video. Through extensive experiments, we show that our joint modeling substantially improves both spatial and temporal super-resolution and sets a new state of the art for multiple benchmarks: across a wide range of upscaling factors, it delivers sharper and temporally more consistent reconstructions than existing baselines, while being computationally more efficient. Project page: https://v3vsr.github.io.