The high--precision HERA data allow searches for up to TeV scales ``Beyond the Standard Model'' contributions to electron--quark scattering. 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.