The noun steeplechase denotes:
– a horse race run on a racecourse having ditches and hedges as jumps;
– a running race in which runners must clear hurdles and water jumps.

ORIGIN

This noun originally designated a horse race across a stretch of open countryside, with a church steeple in view as goal; this was explained in the paragraph where appears the first known instance of steeplechase, from The Sporting Magazine; or, Monthly Calendar of the Transactions of the Turf, the Chase, and every other Diversion Interesting to the Man of Pleasure and Enterprize (London) of April 1793:

The Hon. Mr. O’Hea and Captain Magrath ran a steeple-chase near Galloway, in Scotland, lately, for a bet of fifty guineas, which was won by the latter, after a hard contest.—To some of our readers it may perhaps be necessary to say, that this amusement consists of riding over hedge and ditch as fast as possible, towards the nearest steeple from the place of starting.

In its first edition (Oxford, 1919), A New English Dictionary on Historical Principles (as the Oxford English Dictionary was originally called) recorded the phrase to hunt the steeple, used in The Edinburgh Advertiser (Scotland) of 15^th April 1785:

His Lordship and another gentleman determined to hunt the steeple. This is a common amusement among people of fashion, and consists in the horsemen riding helter skelter towards the first steeple that may catch their eye, and he that is first in is the best man.

This gave rise to compounds with hunt instead of chase, as in Observations, relative chiefly to picturesque Beauty, made in the Year 1772, on several Parts of England; particularly the Mountains, and Lakes of Cumberland, and Westmoreland (London, 1786), by William Gilpin (1724-1804), English schoolmaster and writer on art:

From Leicester the country still continues flat and woody; stretching out into meadows, pastures, and common fields. The horizon, on every side, is generally terminated by spires. Oftener than once we were able to count six, or seven adorning the limits of one circular view.
Of all the countries in England, this is the place for that noble species of diversion, to which the inventive genius of our young sportsmen hath given the name of steeple-hunting. In a dearth of game, the chassieurs [misprint for chasseurs] draw up in a body, and pointing to some conspicuous steeple, set off, in full speed towards it, over hedge and ditch. He who is so happy, as to arrive first, receives equal honour, it is said, as if he had come in foremost, at the death of the fox.

According to the veterinary surgeon and author William Taplin (died 1807) in The Sporting Dictionary, and Rural Repository of General Information upon every Subject appertaining to the Sports of the Field (London, 1803), steeple chase was often synonymous with train scent, or drag, denoting a chase using an object dragged along the ground to make a scent for hounds to follow [cf. footnote]:

A hunting match (generally termed a steeple chase) is made by parties, to ride their own horses across a country to some point agreed on, encountering all difficulties, and taking the leaps in stroke: this kind of match is, upon most occasions, run with a few couple of hounds; a person going forward with a drag to the spot appointed where the match is to be decided.
[…]
Wild-goose chase,—is neither more or less than a metaphorical allusion to the uncertainty of its termination. This originated in a kind of chase (more properly match) formerly decided in the following way. Two horses having started at the place appointed, continued to rate by the side of each other, till one having obtained the lead, was entitled to proceed in whatever direction the rider pleased, (either by shortening or prolonging the distance to the winning spot previously agreed on,) according to the qualifications of his horse. This kind of chase so frequently terminated in tired or spoiled horses, without a decision, that it was long since changed to a train scent, (that is, a drag across the country;) better known by the denomination of a steeple chase.

Note: train scent, or drag, as defined by William Taplin in the same dictionary:

Train scents,—formerly so called, but now more frequently termed drags, are means by which young hounds may be first entered with old hounds; a body of hounds exercised upon heaths or commons, soon after dawn of day in the summer season: or bets may be decided upon the speed of either hounds or horses, by means of such drag or train scent. They are of different kinds, and very few hounds will refuse to hunt them: when the scent lies well, the wind is still, and the atmosphere free from variation by storms or rain, they will carry it breast high. The skin of hare or fox, newly killed; a slice of bacon, and a red herring firmly united; or either, plentifully impregnated with oil of aniseed; will lead hounds in full cry across any country over which the drag is directed.