What is the meaning of "hypos" here in this passage from Moby Dick?

There are different meanings, plus hypo and hypo- are two different things. (If you want I'll be more specific here, but I avoided it because it wasn't what you asked.)

But the meaning you need is: "Morbid depression of spirits." (Taken from the OED.)

It's the only one that fits in that context, if you read again the passage, it will be much more understandable now.

EDIT: I forgot to mention. This meaning I gave to you is signalled as Obsolete and it says it's an "Abbreviation of hypochondria".

Following up on Alenanno's accurate answer, I note the following entry for "hyps (or hypo)" in J. S. Farmer & W. E. Henley, Slang and Its Analogues Past and Present, volume 3 (1893):

HYPS (or HYPO), subs. (old.).—The BLUE DEVILS [elsewhere defined as "Dejection; lowness of spirits; hypochondria"]. [Citations:] 1710. Swift, Tatler, No, 230. Will Hazard has got the HIPPS, having lost to the tune of five hund'rd pound. [Second citation:] 1729. Swift, Poems (Chalmers, English Poets, 1810, xi., 486). And the doctor was plaguily DOWN IN THE HIPS. 1738. Swift's Polite Conversation, Dial. 1. Her ladyship was plaguily bamb'd. I warrant it put her into the HIPPS. 1811. Lexicon Balatronicum, s.v. 1830. C. Lamb, Pawnbroker's Daughter, i., 2. The drops so like to tears did drip, They gave my infant nerves the HYP. 1854. Haliburton, Americans at Home, i., 176. The old man would give up to the HYPO, and keep his bed for weeks. During this time, he wouldn't say a word, but 'I'm not long for this world.'

The citation from 1811 is to Frances Grose & a Member of the Whip Club, Lexicon Balatronicum: A Dictionary of Buckish Slang, University Wit, and Pick-Pocket Eloquence (1811), which contains this entry:

HYP. The hypochondriac ; low spirits. He is hypped ; he has got the blue devils, &c.

The earliest example by Swift of the three that Farmer & Henley cites may indeed have appeared very soon after the slang term emerged, since it comes in the context of a letter chock-a-block with abbreviated words and other bits of slang that Swift was mocking (and evidently detested). For context, here is a bit more from The Tatler No. 230 (September 28, 1710), presented as a letter to Isaac Bickerstaff {Swift] from a gentleman of some refinement who is passing along a note that he supposedly had recently received from someone acquainted with the very latest slang:

These two Evils, Ignorance, and want of Taste, have produced a third ; I mean the continual Corruption of our English Tongue, which, without some timely Remedy, will suffer more by the false Refinements of twenty Years past, than it hath been improved in the foregoing hundred. ...

But instead of giving you a List of the late Refinements crept into our Language, I here send you the Copy of a Letter I received some Time ago from a most accomplished Person in this Way of Writing ; upon which I shall make some Remarks. It is in these Terms:

Sir, I cou'd n't get the Things you sent for all about Town——I thôt to ha come down my self, and then I'd brôt 'um ; but I ha'nt don't, and I believe I can't do't that's Pozz—Tom begins to g'imself Airs, because he's going with the Plenipo's——'Tis said the French King will bamboozl' us agen, which causes many Speculations. The Jacks and others of that Kidney are very uppish, and alert upon't, as you may see by their Phizz's———Will Hazard has got the Hipps, having lost to the Tune of five hundr'd Pound, thô he understands Play very well, no Body better. He has promis't me upon Rep, to leave off Play, but you know 'tis a weakness he's too apt to give into, thô he has as much Wit as any man, no body more. He has lain incog ever since——

The author of the letter to Bickerstaff goes on to identify and condemn the source of short forms such as "Hipps":

And this ["natural Tendency toward Barbarity, which delights in Monosyllables"] is still more visible in the next Refinement, which consists in pronouncing the first Syllable in a Word that has many, and dismissing the rest, such as Phizz, Hipps, Mobb, Pozz, Rep, and many more, when we are already overloaded with Monosyllables, which are the Disgrace of our Language.

The use of Hipps for hypochondria thus seems to have been new enough in 1710 to strike readers as outlandish, just as Phizz, Pozz, Plenipo, and incog would have done.

The earliest instance I could find of the spelling hypos for the short form of hypochondria is from "A Cure for the Hyp," in The London Magazine (November 1732):

Mr. Spectator says, The old Distemper, called Melancholy, was happily exchanged for the Vapours, and afterwards for the Hyp, till at last it took up the now current Appellation of the Spleen ; tho' a learned Doctor of the West has written a Tract, in which he divides the Spleen and Vapours, not only into the Hyp, the Hypos, and the Hypocons, but subdivides these Divisions into the Mark-ambles, the Moon-palls, the Strong-Fives, and the Hockogrokles. I am almost sure I am right, thio' I have mislaid the Book among my Romances.

Conclusions

There seems to be no disagreement among knowledgeable sources that hypo and hypos (and hyp and hyps and hipps) arose as slang short forms of hypochondria and/or hypochondriac. The slang shortening was in use by 1710, but perhaps not much earlier than that. The particular spelling hypos was in use (although by no mean to the exclusion of other spellings) by 1732.