Abstract: We present SIMBIG, a likelihood-free inference framework for analyzing galaxy clustering using a fully simulation-based approach. We apply SIMBIG to the BOSS CMASS galaxy sample using an $N$-body simulation-based forward model that includes a flexible galaxy-halo model, detailed survey geometry, and realistic observational systematics. As demonstration and validation, we use SIMBIG to analyze the galaxy power spectrum out to $k_{\rm max} = 0.5\,h/{\rm Mpc}$. We derive constraints on $\Omega_m$ and $\sigma_8$ that are a factor of 1.1 and 3, respectively, tighter than previous results. This improvement comes from the extra cosmological information available on non-linear scales that we can extract with our simulation-based approach. Furthermore, we use a suite of test simulations to confirm that our LFI approach produces conservative estimates of the true posterior. In subsequent work, we will apply SIMBIG to analyze higher-order statistics and non-standard observables such as the bispectrum, marked power spectrum, and wavelet scattering-like statistics.