Holographic multiple-input multiple-output (HMIMO) communication systems utilize spatially-constrained arrays equipped with large number of antennas (NoA) to benefit from increased spatial multiplexing and spatial resolution gains. We consider a multi-user HMIMO system under an electromagnetic wave compliant channel model, and study its energy-efficiency (EE). We first derive closed-form expressions of the ergodic achievable rates under maximum ratio transmission (MRT) and zero-forcing (ZF) precoding in the downlink, and maximum ratio combining (MRC) and ZF combining in the uplink, implemented at the base station (BS) with reduced complexity dictated by the number of degrees-of-freedom (DoF) offered by the channel. Using these expressions, we formulate an EE maximization problem with respect to the power allocation (PA) and the NoA arranged within spatially-constrained HMIMO surfaces at the BS and users, and solve it using an alternating optimization algorithm. For fixed PA, the optimal NoA is derived as the solution of two analytical equations, while for fixed NoA we use sequential fractional programming to obtain the optimal PA. Numerical results yield useful insights into the EE performance in different operating regimes and under different side-lengths of HMIMO surfaces. The presented results show that under MRT and MRC in the downlink and uplink respectively, deploying more antennas in excess of the number of DoF increases the EE in low power budget (noise-limited) regime, whereas fixing the NoA to the number of DoF maximizes the EE in higher power budget (interference-limited) regime. On the other hand, under ZF precoders and combiners, the NoA that achieve the optimal EE is larger than the DoF under all power budgets, with the number of additional antennas decreasing with increasing power budget.