Distributed acoustic sensing (DAS) is regarded as a novel acquisition technology for seismic data. Compared with conventional electrical geophones, DAS has a series of obvious advantages including low-cost, high spatial resolution, good coverage, and strong resistance to the harsh environment. Noise attenuation is an essential step in seismic data processing. However, there are two main difficulties faced by the denoising task of DAS seismic data. On the one hand, some background noise in DAS seismic data, such as optical low-frequency noise, horizontal noise, and fading noise, is unique and not presented in the conventional seismic data; on the other hand, the signal-to-noise ratio (SNR) of DAS seismic data is relatively low. Recently, a convolutional neural network (CNN) has shown superior denoising performance compared to the traditional method. To follow this promising trend, we propose a multi-scale interactive convolutional neural network (MSI-Net) and apply it to denoise the challenging DAS seismic data. Different from most of the existing CNN architecture used in seismic data denoising, the MSI-Net considers both coarse-scale and fine-scale features by improving the inherent serial convolution to multi-scale parallel convolution, which is beneficial to recover detailed information. Moreover, we utilize some connections to achieve the information interaction between different scales, which promotes the flow of information and enables the network to extract more informative multi-scale features from the DAS seismic data. Moreover, both synthetic and real examples demonstrate that the proposed MSI-Net can effectively attenuate a variety of unique DAS background noise and also completely recover the weak signals. Compared with conventional CNN architecture, MSI-Net exhibits better performance in global SNR and local details.