The phrase Black Hole of Calcutta (also, in early use, Black Hole at Calcutta) denotes an oppressive, very confined or crowded space.

This phrase occurs, for example, in Oh, what a summer!, by Clive Aslet, published in The Daily Telegraph (London, England) of Tuesday 3^rd September 2013:

Going between appointments in London yesterday, it was so nice that I walked. So what if I were late? In the dolce far niente mindset that the heat induces, nobody seemed to mind. Admittedly, what strikes the lunch-time flaneur as strolling temperature can present a different face to hard-pressed commuters, squeezed on to public transport during the rush hour, in conditions that would make the Black Hole of Calcutta seem pampering. Their compensation comes later, when their spirits revive as they enjoy balmy evenings on the lawn, with a long drink to hand.

The term black hole denotes a place of confinement as punishment. The following, for example, is from the Stamford Mercury: Being Historical and Political Observations on the Transactions of Europe, together with Remarks on Trade (Stamford, Lincolnshire, England) of Thursday 18^th June 1724:

On the 5^th Instant, one Creed, a Drummer, was conveyed to the Tower from the Savoy, under a Guard of five Musketeers, being Hand cuff’d; he having made his Escape out of the Black Hole some time since, where he was confined with two others for certain Misdemeanors. He is sentenced thither again till farther Orders.

The phrase Black Hole of, or at, Calcutta refers specifically to the punishment cell at the barracks in Fort William, Calcutta, Bengal, in which, on Saturday 19^th June 1756, after the fall of the fort, 64 British and Anglo-Indian soldiers and civilians were reputedly confined overnight in crowded conditions, only 21 surviving until the morning. The following, for example, is the title of a story published in The Gentleman’s Magazine, and Historical Chronicle (London, England) of February 1758:

A Genuine Narrative of the Sufferings of the Persons who were confined in the Prison called the Black Hole, in Fort William at Calcutta, in the Kingdom of Bengal, after the Surrender of that Place to the Indians in June 1756, from a Letter of J. Z. Holwell, Esq; to William Davis, Esq.

The earliest occurrences that I have found of the phrase Black Hole of, or at, Calcutta used to denote an oppressive, very confined or crowded space are as follows, in chronological order:

1-: From Extracts of a letter from Lisbon, Feb. 14, published in The Caledonian Mercury (Edinburgh, Midlothian, Scotland) of Wednesday 21^st March 1764:

This day the workmen began to erect scaffolding at Merchant Taylors hall, for accommodating the general meeting of the East India proprietors, which will be held there next Wednesday.
The reason why the proprietors have their next meeting at Merchant Taylors hall, is because the rooms both at the East India and South-sea houses, where the last meetings were held, were much too small for the great number of persons who attended.
Several gentlemen have caught great colds by attending the two last meetings, and declared they felt so much heat when within the rooms, that they esteemed themselves to have been confined in the black hole at Calcutta.

2-: From The Bath and Bristol Chronicle (Bath, Somerset, England) of Thursday 6^th July 1769:

A letter from Cambridge dated July 2, says,—
“Yesterday was the installation; ’tis amazing what a concourse of company attended at the Senate House from nine till half past ten; when, through very bad management, the gate being opened, the push to get in was so violent, no order being observed, that the ladies and gentlemen were wedged so close together as to beggar description; caps, laces, ruffles, jewels, &c. were trampled upon; and when the house was full, it resembled another black-hole at Calcutta: you must imagine many fainted away; and the only real damage (for bruises, &c. must be excluded) was a lady’s arm being broke.

3-: From Saunders’s News-Letter, and Daily Advertiser (Dublin, County Dublin, Ireland) of Monday 27^th September 1784:

The public will be astonished to hear, that in the midst of our endeavours to reform the abuses of India, the European inhabitants of Bengal have taken up the idea of reforming themselves. We have been divided in our scheme, but theirs is so harmless as well as so original, that the great men are mightily entertained with the thought.—The plan is neither more nor less than to build a Church in Calcutta for the worship of God—and this is an original idea. It is one of the most populous and opulent cities in the universe. It is superior to London, or even to Paris, in its debaucheries.—They have their play-house, their gaming-houses, masquerades, their assemblies, horse-races, their brothels, and their pleasures of all kinds; but it is now only that they have thought of a Church. We may be pretty confident that this Church will never become like the Black Hole of Calcutta, dangerous from being over crowded.

4-: From a paragraph of a letter from a gentleman in Jamaica to a friend in Edinburgh, dated Tuesday 15^th November 1803—published in the Edinburgh Weekly Journal (Edinburgh, Midlothian, Scotland) of Wednesday 18^th January 1804:

“Before I quit the subject of St Domingo, I will give you an anecdote which I have from good authority, and which will give you an additional liking for the new French principles of morality, republicanism, and fraternization. During the late struggle in the island, Le Clerc, and after him Rochambeau, employed in the harbours of Port-au-Prince and Cape Francois, several vessels, into the holds of which were crammed by force, and indiscriminately, blacks, mulattoes, and whites, who were unfriendly to the cause of Bonaparte, and whom it was expedient to get rid of.—Those poor devils, after undergoing the miseries of the Black Hole at Calcutta, were carried some leagues from land, and, at the point of the bayonet, without mercy, pushed into the ocean one after another. These vessels were very appropriately called Stiflers.”