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 data from recent CMB, BAO and H_0 measurements. We focus on possible variations of the fine structure constant alpha and the electron mass m_e in the early universe, and study the degeneracies between these constants and other cosmological parameters, such as the dark energy equation of state parameter w_DE, the massive neutrinos fraction f_ u, the effective number of relativistic species N_eff, and the primordial helium abundance Y_He. When only one of the fundamental constants is varied, our final bounds are alpha / alpha_0 = 0.9957_{-0.0042}^{+0.0041} and m_e /(m_e)_0 = 1.006_{-0.013}^{+0.014}. For their joint variation, our results are alpha / alpha_0 = 0.9901_{-0.0054}^{+0.0055} and m_e /(m_e)_0 = 1.028 +/- 0.019. Although when m_e is allowed to vary our constraints on w_DE are consistent with a cosmological constant, when alpha is treated as a free parameter we find w_DE = -1.20 +/- 0.13; more than 1 sigma away from its standard value. When f_ u and alpha are allowed to vary simultaneously, we find f_ u < 0.043 (95% CL), implying a limit of sum m_ u < 0.46 eV (95% CL), while for m_e variation, we obtain f_nu < 0.086 (95% CL), which implies sum m_ u < 1.1 eV (95% CL). When N_eff or Y_He are considered as free parameters, their simultaneous variation with alpha provides constraints close to their standard values (when the H_0 prior is not included in the analysis), while when m_e is allowed to vary, their preferred values are significantly higher. In all cases, our results are consistent with no variations of alpha or m_e at the 1 or 2 sigma level.
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