The process of e+e-→pp̄ is studied at 22 center-of-mass energy points (s) from 2.00 to 3.08 GeV, exploiting 688.5 pb-1 of data collected with the BESIII detector operating at the BEPCII collider. The Born cross section (σpp̄) of e+e-→pp̄ is measured with the energy-scan technique and it is found to be consistent with previously published data, but with much improved accuracy. In addition, the electromagnetic form-factor ratio (|GE/GM|) and the value of the effective (|Geff|), electric (|GE|), and magnetic (|GM|) form factors are measured by studying the helicity angle of the proton at 16 center-of-mass energy points. |GE/GM| and |GM| are determined with high accuracy, providing uncertainties comparable to data in the spacelike region, and |GE| is measured for the first time. We reach unprecedented accuracy, and precision results in the timelike region provide information to improve our understanding of the proton inner structure and to test theoretical models which depend on nonperturbative quantum chromodynamics.