Grant-free non-orthogonal multiple access is an emerging research topic in machine-type communications, which is used to reduce signaling overhead. In this context, this letter introduces a novel joint channel estimation (CE) and multiuser detection (MUD) framework for the frame based multi-user transmission scenario where users are (in)active for the duration of a frame. First, considering the inherent frame-wise joint sparsity of the pilot and data phases in the entire frame, we formulate the multiple measurement vector-compressive sensing (MMV-CS) framework. Then, transfer the MMV-CS to a block-sparse single measurement vector-CS (BS-SMV-CS) model. Finally, to make explicit use of the block sparsity inherent in the BS-SMV-CS model and consider that the user sparsity level should be unknown for receiver, an enhanced subspace pursuit (SP) algorithm is developed, i.e., block sparsity adaptive SP. Superior performance of the proposed joint CE and MUD framework is demonstrated by simulation results.