The broadcast ionospheric model is one of the main methods for eliminating ionospheric delay errors for the Global Navigation Satellite Systems (GNSS) single-frequency users. GPS Klobuchar model (GPSK8) is the widely used broadcast ionospheric model for GPS, while BDS usually implements the BDS Klobuchar model (BDSK8) and BeiDou Global Broadcast Ionospheric Delay Correction Model (BDGIM). Geomagnetic storms may cause interference within the ionosphere and near-Earth space, compromising the accuracy of ionospheric models and adversely affecting the navigation satellite systems. This paper analyzes the static Standard Point Positioning (SPP) accuracy of GPS and BDS by implementing the broadcast ionospheric models and then investigates the impact of strong geomagnetic storms occurring in 2021 on positioning accuracy. The results show that the global 3D positioning accuracy (95%) of GPS + GPSK8, BDS + BDSK8, and BDS + BDGIM are 3.92 m, 4.63 m, and 3.50 m respectively. BDS has a better positioning accuracy in the northern hemisphere than that of the southern hemisphere, while the opposite is valid for GPS. In the mid-latitude region of the northern hemisphere, BDS + BDSK8 and BDS + BDGIM have similar positioning accuracy and are both better than GPS + GPSK8. The positioning accuracy after applying those three broadcast ionospheric models shows the superior performances of winter and summer over spring and autumn (based on the northern hemisphere seasons). With the exception of during winter, nighttime accuracy is better than that of daytime. The strong geomagnetic storm that occurred on the day of year (DOY) 132, 2021 has an impact on the positioning accuracy for only a small number of stations; however, the global average positioning accuracy is not significantly affected. The strong geomagnetic storms that occurred in DOY 307 and DOY 308 have a significant impact on the positioning accuracy of dozens of stations, and the global average positioning accuracy is affected to a certain extent, with some stations experiencing a serious loss of accuracy. Decreased degrees in positioning accuracy is proportional to the intensity of the geomagnetic storm. Of the 33 IGS Multi-GNSS Experiment (MGEX) stations worldwide, those located in the low and mid-latitudes are more significantly affected by the geomagnetic storms compared with higher latitudes. Evident fluctuations of the positioning errors existed during the strong geomagnetic storms, with an increase in extreme values, particularly in the up direction.