We present a detailed model atmosphere analysis of two of the oldest stars known in the solar neighbourhood, the high proper motion white dwarfs SDSS J110217.48+411315.4 (hereafter J1102) and WD 0346+246 (hereafter WD0346). We present trigonometric parallax observations of J1102, which places it at a distance of only 33.7 ± 2.0 pc. Based on the state-of-the-art model atmospheres, optical, near-infrared, mid-infrared photometry and distances, we constrain the temperatures, atmospheric compositions, masses and ages for both stars. J1102 is an 11-Gyr-old (white dwarf plus main-sequence age), 0.62 M⊙ white dwarf with a pure H atmosphere and Teff= 3830 K. WD0346 is an 11.5-Gyr-old, 0.77 M⊙ white dwarf with a mixed H/He atmosphere and Teff= 3650 K. Both stars display halo kinematics and their ages agree remarkably well with the ages of the nearest globular clusters, M4 and NGC 6397. J1102 and WD0346 are the closest examples of the oldest halo stars that we know of.