Solution 1:

Today—in U.S. English, at least—beef is used to refer to meat from various breeds of what Merriam-Webster's Eleventh Collegiate Dictionary (2003) calls "bovine animals" (cows or oxen). It is thus on a par with mutton (sheep meat), pork (pig meat), and venison (deer meat). All of these meats are interpreted collectively, so there is never an occasion to refer to these meats as muttons, porks, venisons, or beefs.

But in the past, beeves was often used to refer to the living animals that yielded beef. For example, from a footnote in The Natchez Court Records, 1767–1805: Abstracts of Early Records (1979):

John Bisland petitions that during the time of the revolt, Alexander Grading did kill and cause to be killed a number of beeves for the use of himself and company which said beeves were the property of your petitioner for which Mr. Grading had no order from his commander, Mr. Blommart, as Mr. Blommart afterwards declared upon his word of honor. Mr. Grading them before setting off for the Chickasaw Nation offered your petitioner a note of hand for a part of the beeves in the present of Mr. Spell... April 18, 1782 // To be communicated, To Alexander Grading debtor to John Bisland. Six beeves which you killed at $20. ... Deposition of Sterling Spell that he was present when Grading killed two of the beeves and also when Grading said he would pay for them. // Parties and witnesses heard, we do condemn Alexander Grading to the payment of $80 in month of December next, being the value of four beeves killed by him belonging to Mr. Bisland, during the revolt. Sig: Grand-Pre.

And in an 1866 diary of a longhorn cattle drive from Texas northward, in Paul Wellman, The Trampling Herd: The Story of the Cattle Range in America (1988):

Next day the diary noted "Hunt cattle again Men all tired & want to leave. am in the Indian country am annoyed by them believe they scare the Cattle to get pay to collect them. ... Two men and Bunch Beeves lost— Horses all give out & Men refused to do anything." And on the succeeding day: "Hard rain & wind Storm Beeves ran & had to be on Horse back all night. Awful night, wet all night clear bright morning. Men still lost quit the Beeves and go Hunting Men is the word—4 p.m. Found our men with Indian guide & 195 Beeves 14 miles from camp. allmost starved not having had a bite to eat for 60 hours got to camp about 12 m Tired."

The second example is especially interesting because the diarist uses the term beeves and the term cattle in slightly different ways: He uses cattle (twice) when referring generically to the herd being driven or the portion of it that the cowboys need to hunt after they stampede. But beeves seems to refer to countable animals that are intended to be sold for slaughter at the end of the drive.

Another notable thing about both excerpts is that, despite multiple mentions of beeves, they never use the term beef (singular) to mean a live cow or ox. That does happen, however, in Laban S. Records, Cherokee Outlet Cowboy (1937):

Frank Bates was friendly with all of us, but did not enjoy camping out. He liked to be with when we were cutting out beeves. On one occasion he hitched Bob and Dirk, his favorite horses, to his top buggy an drove out where we were at work. He motioned to me, "Don't you think I could drive Bob and Dirk into the herd and pick out a beef just as good as you?"

"Frank you know what would happen if you made a dash at a steer with the team and buggy; you'd have a stampede!" I replied.

Here, beef and beeves seem to be used for animals about to be dispatched, while steer refers to any (castrated male) animal in the herd (steers were the animals usually involved in cattle drives). Elsewhere the author refers to the entire herd as the herd or the beef herd.

Webster's Dictionary of the English Language (1852) goes into some detail about how the term beef was used in the middle of the 19th century in the United States:

BEEF, n. 1. An animal of the bovine genus, whether ox, bull, or cow ; but used of those which are full grown or nearly so. In this, which is the original sense, the word has a plural, beeves. 2. The flesh of an ox, bull, or cow, or of bovine animals generally, when killed. In popular language, the word is often applied to the live animal, as, an ox is good beef ; that is, well fattened. In this sense, the word has no plural.

Solution 2:

beeves (n.)
original plural of beef (compare boevz, plural of Old French buef), now only in restricted use.

beef (n.)
c.1300, from Old French buef "ox; beef; ox hide" (11c., Modern French boeuf), from Latin bovem (nominative bos, genitive bovis) "ox, cow," from PIE root *gwou- "cow, ox, bull" (see cow (n.)).

Original plural was beeves.

From the Online Etymology Dictionary

Essentially, as has been said numerous times already, we wouldn't really use it anymore.

Here's a source describing beeves as referring to several different kinds of beef.

The meat from the Ox? That’s beef. The meat from the Bull? Why, we call that beef, too.

You can extend this explanation to include the animals themselves. Say I have several oxen and bulls I'm raising for meat. I have quite a few beeves.

At any rate, very interesting question. I never knew that word existed before today.

Solution 3:

  • We sell our grassfed beeves by live weight.

  • Half and Whole Beeves Information

Both lines above are from the Mother Earth Meats website.

This one is from the Unused Words website.

If, however you order the selection of beeves, you, my friend, are a fully-fledged gourmand.

Solution 4:

I have heard cattle ranchers use this word, mostly out of a mild frustration with the lack of a common, gender-neutral word for cow/bull.

Not that they are PC. For some purposes, cows and bulls are very different (a herd of 100 cows might have only a single bull), but sometimes, you need a word for them collectively. For example, "That new alfalfa is not as popular with my __."

So they say beeves, more or less jocularly.