Old English stepel (Mercian), stiepel (West Saxon) "high tower," related to steap "high, lofty," from Proto-Germanic *staupilaz (see steep (adj.)). Also the name of a lofty style of women's head-dress from the 14th century. Steeple-house (1640s) was the old Quaker way of referring to "a church edifice," to avoid in that sense using church, which had with them a more restricted meaning.
c. 1300, chacen "to hunt; to cause to go away; put to flight," from Old French chacier "to hunt, ride swiftly, strive for" (12c., Modern French chasser), from Vulgar Latin *captiare "try to seize, chase" (source of Italian cacciare, Catalan casar, Spanish cazar, Portuguese caçar "to chase, hunt"), from Latin captare "to take, hold," frequentative of capere "to take, hold" (from PIE root *kap- "to grasp"). The Old French word is a variant of cacier, cachier, making chase a doublet of catch (v.).
The meaning "run after" for any purpose is by mid-14c. Related: Chased; chasing. Ancient European words for "pursue" often also cover "persecute" (Greek diōkō, Old English ehtan), and in Middle English chase also meant "to persecute." Many modern "chase" words often derive from verbs used primarily for the hunting of animals.
<a href="https://www.etymonline.com/word/steeplechase">Etymology of steeplechase by etymonline</a>
Harper, D. (n.d.). Etymology of steeplechase. Online Etymology Dictionary. Retrieved $(datetime), from https://www.etymonline.com/word/steeplechase