The high-precision HERA data allow searches for "beyond the Standard Model" contributions to electron-quark scattering up to TeV scales. Combined H1 and ZEUS measurements of inclusive deep inelastic cross sections in neutral and charged current $ep$ scattering are considered, corresponding to a luminosity of around 1 $fb^{{} - 1}$. A new approach to the beyond the Standard Model analysis of the inclusive $ep$ data is presented; simultaneous fits of parton distribution functions and contributions of "new physics" processes are performed. Considered are possible deviations from the Standard Model due to a finite radius of quarks, described within the quark form-factor model, and due to new electron-quark interactions in the framework of $eeqq$ contact interactions (CI). The resulting 95% C.L. upper limit on the effective quark radius is $0.43 \cdot 10^{{} - 16}$ cm. The limits on the CI mass scale extend up to 10 TeV depending on the CI scenario.