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The phrase (as) thick as herrings is said of a great number of persons or things, especially when pressed against one another.
—Synonyms: like herrings in a barrel and to be packed like sardines.

This phrase (as) thick as herrings occurred, for example, in the following from Floating Death in the Sea, by Captain G. P. Phillips, former Commodore of the Clan Line, published in the Bucks Examiner (Chesham, Buckinghamshire, England) of Friday 15^th December 1939 [page 9, column 4]:

During the last war I had to carry a cargo of dynamite out to Bombay. You can guess what I thought of the job, with submarines as thick as herrings all through the Mediterranean, and mines floating about like seaweed everywhere! But I got through safely.

Contrary to what the current online edition of the Oxford English Dictionary states, the phrase (as) thick as herrings does not refer to herrings “in shoals” [sic], but to herrings in a barrel—cf., below, quotation 1.

These are, in chronological order, the earliest occurrences of the phrase (as) thick as herrings that I have found:
—Note: Quotations 2, 3 & 4 below are from works by the English satirist John Wolcot (1738-1819), published under the pen name of Peter Pindar:

1-: From Nouveau Dictionnaire François-Anglois, et Anglois-François. De M^r. A. Boyer (Paris: C. Panckoucke – Amsterdam & Leipzig: Arkstée & Merkus – Rotterdam: H. Beman – 1776), by Louis Chambaud and J. B. Robinet [tome 1, page 379, column 1, s.v. hareng]:

Rangé, pressé comme des harengs en caque (se dit de personnes ou de choses rangées & pressées l’une contre l’autre) Stowed as thick as herrings.
     translation:
Stowed, pressed like herrings in barrel (is said of persons or of things stowed & pressed against one another) Stowed as thick as herrings.

2-: From The Soldier and the Virgin Mary. A Tale, published in Subjects for Painters (London: Printed for G. Kearsley, 1789), by Peter Pindar [page 52]:

A Soldier at Loretto’s wond’rous chapel,
To parry from his soul the wrath divine,
That follow’d mother Eve’s unlucky apple,
Did visit oft the Virgin Mary’s shrine;
Who ev’ry day is gorgeously deck’d out,
In silks or velvets, jewels, great and small […].
[…]
One day as he was making love and praying,
And pious Aves, thick as herrings, saying,
And sins so manifold confessing;
He drew, as if to whisper, very near,
And twitch’d a pretty diamond from her ear,
Instead of taking the good lady’s blessing.

3-: From Celebration, or, The Academic Procession to St. James’s; An Ode (London: Printed for John Walker, 1794), by Peter Pindar [page 12]:

See! from yon Dome, amid th’ expectant throng,
Slow moves the tribe of Benjamin along,
While Fame before them with her trumpet flies;
Whilst on their heads, from bulks and chimney-tops,
As thick as herrings, or as thick as hops,
Wild Admiration casts her countless eyes.

4-: From Ode to the Lion Ship of War, on her Return with the Embassy from China, published in Pindariana; or Peter’s Portfolio (London: Printed by T. Spilsbury and Son, for J. Walker [&c.], 1794), by Peter Pindar [page 226]:

Kings think they hold the world’s esteem;
Think, too, the conscience sound, though full of holes,
And virtues, thick as herrings, in their souls.

5-: From The Humours of Bunmahon, a Country Bathing place, published in Miscellaneous Poems (Waterford: Printed by John Veacock, 1798), by Edward M. Mandeville [page 39]:

The owners on their half-doors leaning,
Impatient for that pleasing sign,
When they may sit as well as dine.
The signal made that dinner’s done,
And all who’re bidden now may run;
Quick from their hives they sally forth
As thick as herrings from the North.