The use of the work "obelisk" is most likely a malapropism here. The narrator is confusing the less familiar word basilisk with the more familiar word obelisk.
An obelisk is a type of monument and won't cast any glances at anyone. The basilisk, by contrast,
> is a legendary reptile reputed to be a serpent king who can cause death with a single glance.
It only exists in myths and fables, hence the phrase "fabled basilisk". In English literature, the basilisk somehow got confused with the cockatrice. You can find several instances of "cockatrice" in Shakespeare's works, where it is typically associated with a deadly glance.