  Using a new recently compiled milliarcsecond compact radio data set of 120
intermediate-luminosity quasars in the redshift range $0.46< z <2.76$, whose
statistical linear sizes show negligible dependence on redshifts and intrinsic
luminosity and thus represent standard rulers in cosmology, we constrain three
viable and most popular $f(T)$ gravity models, where $T$ is the torsion scalar
in teleparallel gravity. Our analysis reveals that constraining power of the
quasars data (N=120) is comparable to the Union2.1 SN Ia data (N=580) for all
three $f(T)$ models. Together with other standard ruler probes such as Cosmic
Microwave Background and Baryon Acoustic Oscillation distance measurements, the
present value of the matter density parameter $\Omega_m$ obtained by quasars is
much lager than that derived from other observations. For two of the models
considered ($f_1$CDM and $f_2$CDM) a small but noticeable deviation from
$\Lambda$CDM cosmology is present, while in the framework of $f_3$CDM the
effective equation of state may cross the phantom divide line at lower
redshifts. These results indicate that intermediate-luminosity quasars could
provide an effective observational probe comparable to SN Ia at much higher
redsifts, and $f(T)$ gravity is a reasonable candidate for the modified gravity
theory.
