We examine the ground state and the excitations of a one-dimensional Heisenberg spin-1/2 antiferromagnet with alternating dimers and four-spin plaquettes (a dimer-plaquette chain). The properties of the system depend on the competing dimer and plaquette bonds. Several exact, exact numerical and perturbational results are presented. We find that the system is gapped for all parameter values. The spin pair correlation functions can be characterized by three different correlation lengths for dimer-dimer, dimer-plaquette and plaquette-plaquette correlations. For the latter we find an effective S = 1 Haldane-like behaviour in the limit of dominating dimer bonds. On introducing frustration, the system undergoes a first-order phase transition to a fully dimerized state. As regards the phase relationships of the ground-state wave function, the system represents an example showing exact validity of the Marshall-Peierls sign rule in a strongly frustrated antiferromagnet. The model considered is related to the recently found 1/5-depleted square-lattice Heisenberg system .