Studies of extremely metal-poor stars indicate that chemical abundance ratios [X/Fe] have a root mean square scatter as low as 0.05dex (12%). It remains unclear whether this reflects observational uncertainties or intrinsic astrophysical scatter arising from physical conditions in the ISM at early times. We measure differential chemical abundance ratios in extremely metal-poor stars to investigate the limits of precision and to understand whether cosmic scatter or observational errors are dominant. We used high-resolution (R~95000) and high signal-to-noise (S/N=700 at 5000{AA}) HIRES/Keck spectra to determine high-precision differential abundances between two extremely metal-poor stars through a line-by-line differential approach. We determined stellar parameters for the star G64-37 with respect to the standard star G64-12. We performed EW measurements for the two stars for the lines recognized in both stars and performed spectral synthesis to study the carbon abundances. Results. The differential approach allowed us to obtain errors of {sigma}(Teff)=27K, {sigma}(logg)=0.06dex, {sigma}([Fe/H])=0.02dex and {sigma}(vt)=0.06km/s. We estimated relative chemical abundances with a precision as low as {sigma}([X/Fe])~0.01dex. The small uncertainties demonstrate that there are genuine abundance differences larger than the measurement errors. The observed Li difference cannot be explained by the difference in mass because the less massive star has more Li. It is possible to achieve an abundance precision around ~0.01-0.05dex for extremely metal-poor stars, which opens new windows on the study of the early chemical evolution of the Galaxy.