Heterogeneity is one of the key features that characterizes the future generation of cellular networks, 5G and beyond. However, streaming high-quality bandwidth-hungry multimedia contents over bandwidth-constrained 5G heterogeneous networks (5G HetNets) involves various significant challenges, including long video start time, video start failures, frequent buffering and stalling, and low quality of experience (QoE). Traditional multimedia streaming technologies, however, do not pay attention to either the available network bandwidth or the interaction between content characteristics and resources. To reduce network strain and improve QoE, we propose 'Context-Aware Streaming over 5G HetNets (CASH)' that allows us to achieve a tradeoff between content-context and network-context. The proposed CASH fundamentally works in a multi-step process. First, the CASH comes with an integrated architecture that includes a media server, a flow scheduler, and a single radio controller (SRC). The SRC and the user equipment (UE) of interest cooperatively prepare a metadata file that contains the network-context. Second, based on the metadata file, which can be accessed from the SRC in the media preparation server, we analyze and cluster the contents based on the content-context, e.g., the actual bitrate of each scene. The metadata file is then updated by adding the content-context information. Third, the flow scheduler basically controls the flow of the clusters of the contents in the server-push mode and conveys that to the appropriate radio access technology (RAT) conforming to the bitrate of the clusters and bandwidth delivered by RATs. Finally, the UE will aggregate the received packets and will play-back the content. We analytically show the validity of CASH. Also, extensive simulations are performed to demonstrate that CASH offers substantial performance improvements compared with exiting works in terms of peak data rate, latency, users' experiences, and spectral efficiency.