(52/52 * 3/51) = 1 pocket pair or a, which says you start off with any card in the deck, then multiply by the odds of pairing that card

a * (48/50 * 3/49) = 2 different pocket pairs or b, which says you start off with 1 card different from the first hand (meaning there's 2 that can't hit)

b * (44/48 * 3/47) = 3 different pocket pairs or c, which says that you start off with 1 card different in the first two hands (meaning there's 4 that can't hit)

c * (40/46 * 3/45) = 4 different pocket pairs or d, which says you start off with 1 card different from the first three hands (meaning there's 6 cards in the deck that can't hit)

d * (36/44 * 3/44) = 5 different pocket pairs or e, which says you start off with 1 card different from the first four hands (meaning there's 8 cards in the deck that can't hit)

The total so far is (52*48*44*40*36*3*3*3*3*3)/(52*52*50*49*48*47*46*45*44) or 486/17214925

Next, take the total number of possible combinations the board can have, which is 43*42*41*40*39.

Next, say cards a, b, c, and d must show up on the board, and any other card in the deck that isn't a, b, c, d, or e. Since there's 2 of each card left and 43 cards in the deck, we say the possible number of combinations would then be 2*2*2*2*(43-10).

So the odds of 4/5 hands hitting the board are (2*2*2*2*(43-10))/(43*42*41*40*39) = 11/2406495

And the odds of this situation happening, where the specific cards don't matter, only the fact that everyone has a pocket pair and 4 of them made a set are:

(486/17214925) * (11/2406495) = 1782/13809210312625 or about 1 in 7.75 billion.