We obtain constraints on the variation of the fundamental constants from the full shape of the redshift-space correlation function of a sample of luminous galaxies drawn from the Data Release 9 of the Baryonic Oscillations Spectroscopic Survey. We combine this information with additional data from recent cosmic microwave background, baryon acoustic oscillations and H0 measurements. We focus on possible variations of the fine structure constant α and the electron mass me in the early Universe, and study the degeneracies between these constants and other cosmological parameters, such as the dark energy equation of state parameter wDE, the massive neutrinos fraction fν, the effective number of relativistic species Neff and the primordial helium abundance YHe. In the case when only one of the fundamental constants is varied, our final bounds are α / α _0 = 0.9957_{-0.0042}^{+0.0041} and m_e /(m_e)_0 = 1.006_{-0.013}^{+0.014}. For the joint variation of both fundamental constants, our results are α / α _0 = 0.9901_{-0.0054}^{+0.0055} and me/(me)0 = 1.028 ± 0.019. The variations of α and me from their present values affect the bounds on other cosmological parameters. Although when me is allowed to vary our constraints on wDE are consistent with a cosmological constant, when α is treated as a free parameter we find wDE = -1.20 ± 0.13; more than 1σ away from its standard value. When fν and α are allowed to vary simultaneously, we find fν < 0.043 (95 per cent confidence level, CL), implying a limit of ∑mν < 0.46 eV (95 per cent CL), while for me variation, we obtain fν < 0.086 (95 per cent CL), which implies ∑mν < 1.1 eV (95 per cent CL). When Neff or YHe are considered as free parameters, their simultaneous variation with α provides constraints close to their standard values (when the H0 prior is not included in the analysis), while when me is allowed to vary, their preferred values are significantly higher. In all cases, our results are consistent with no variations of α or me at the 1σ or 2σ level.