We present Very Small Array (VSA) observations (centred on ≈34 GHz) on scales ≈20 arcmin towards a complete, X-ray flux-limited sample of seven clusters at redshift z < 0.1. Four of the clusters have significant Sunyaev–Zel'dovich (SZ) detections in the presence of cosmic microwave background (CMB) primordial anisotropy. For all seven, we use a Bayesian Markov Chain Monte Carlo (MCMC) method for inference from the VSA data, with X-ray priors on cluster positions and temperatures, and radio priors on sources. In this context, the CMB primordial fluctuations are an additional source of Gaussian noise, and are included in the model as a non-diagonal covariance matrix derived from the known angular power spectrum. In addition, we make assumptions of β-model gas distributions and of hydrostatic equilibrium, to evaluate probability densities for the gas mass (Mgas) and total mass (Mr) out to r200, the radius at which the average density enclosed is 200 times the critical density at the redshift of the cluster. This is further than has been done before and close to the classical value for a collapsed cluster. Our combined estimate of the gas fraction (fgas=Mgas/Mr) is 0.08+0.06−0.04 h−1. The random errors are poor (note, however, that the errors are higher than would have been obtained with the usual χ2 method on the same data) but the control of bias is good. We have described the MCMC analysis method specifically in terms of SZ but hope the description will be of more general use. We find that the effects of primordial CMB contamination tend to be similar in the estimates of both Mgas and Mr over the narrow range of angular scales we are dealing with, so that there is little effect of primordials on fgas determination. Using our Mr estimates we find a normalization of the mass–temperature relation based on the profiles from the VSA cluster pressure maps, which is in good agreement with recent M–T determinations from X-ray cluster measurements.