Biomolecular systems like molecular motors or pumps, transcription and translation machinery, and other enzymatic reactions can be described as Markov processes on a suitable network. We show quite generally that in a steady state the dispersion of observables like the number of consumed/produced molecules or the number of steps of a motor is constrained by the thermodynamic cost of generating it. An uncertainty epsilon requires at least a cost of 2k_BT/epsilon^2 independent of the time required to generate the output. We also discuss the relevance of our results in statistical kinetics, which is a field where the aim is to obtain information about the underlying enzymatic scheme by measuring fluctuations in single-molecule experiments. Particularly, we obtain an affinity dependent bound on the Fano factor associated with the number of consumed substrate molecules and we discuss the use of higher order statistical moments to infer properties of the enzymatic scheme.