We propose MESHFREEFLOWNET, a novel deep learning-based super-resolution framework to generate continuous (grid-free) spatio-temporal solutions from the low-resolution inputs. While being computationally efficient, MESHFREEFLOWNET accurately recovers the fine-scale quantities of interest. MESHFREEFLOWNET allows for: (i) the output to be sampled at all spatio-temporal resolutions, (ii) a set of Partial Differential Equation (PDE) constraints to be imposed, and (iii) training on fixed-size inputs on arbitrarily sized spatio-temporal domains owing to its fully convolutional encoder. We empirically study the performance of MESHFREEFLOWNET on the task of super-resolution of turbulent flows in the Rayleigh-Bénard convection problem. Across a diverse set of evaluation metrics, we show that MESHFREEFLOWNET significantly outperforms existing baselines. Furthermore, we provide a large scale implementation of MESHFREEFLOWNET and show that it efficiently scales across large clusters, achieving 96.80% scaling efficiency on up to 128 GPUs and a training time of less than 4 minutes. We provide an opensource implementation of our method that supports arbitrary combinations of PDE constraints.