Fast X-ray transients (FXTs) are flashes of X-rays that last for a few hundreds of seconds to a few hours. An enigmatic population of these transients that did not have a clear origin has been known for several decades, mostly found serendipitously in soft X-ray imaging observations. Recent progress in this field by the Einstein Probe has found that many FXTs are associated with gamma-ray bursts and the collapse of massive stars. Motivated by this, we searched the NuSTAR archive in the harder 3–79 keV band for ∼1000 s duration transients. From 204 Ms of exposure, we present five candidate FXTs, four of which are spectrally hard, with power-law indices −3 < Γ < 0, setting them apart from FXTs discovered in the soft band. Three have potential associations with galaxies at z = 0.1–2, implying 3–79 keV luminosities of 1043–1048 erg s−1 and volumetric event rates of 125–2900 Gpc−3 yr−1. The properties of these NuSTAR FXTs most resemble low-luminosity gamma-ray bursts and would be much more common than their higher-luminosity counterparts in this redshift range.