It was just sitting right there and they seemed OK with it; quite unlike the raging temptation the One Ring is made out to be (especially to the race of men).
They all stayed pretty chill. What gives?
The effect of the Ring on a person isn't instantaneous. Consider that it took Boromir four months (from the Council of Elrond in late October until the breaking of the Fellowship in late February) to be completely overwhelmed by the seduction of the Ring. And even then, he "falls over the brink" only for a relatively brief time, and quickly recovers and realizes that this is only a trick of the Ring, not a rational urge:
For a while he was as still as if his own curse had struck him down; then suddenly he wept. He rose and passed his hand over his eyes, dashing away the tears. 'What have I said?' he cried. 'What have I done? Frodo, Frodo!' he called. 'Come back! A madness took me, but it has passed. Come back!'
(Fellowship of the Ring, Book II, Chapter 10, "The Breaking of the Fellowship")
Keep in mind as well that the Elves of Rivendell may actually have touched the Ring while Frodo was recovering in Rivendell. At any rate, they transferred the Ring from its original chain (by which, as we're told in "The Shadow of the Past", it hung inside his pants pocket) to a new chain around his neck.
With that in mind, it's reasonable to believe that "exposure" to the Ring for even a few days or weeks is not necessarily harmful to an individual.
At the Council of Elrond, Boromir gives us a pretty strong indication that the power of the ring isn't instantaneously corrupting. In fact, to begin with they even dispute that it's Isildur's ring at all, requiring evidence from both Elrond himself and Gandalf before they accept that they need to do anything about it;
Isildur's Bane is found, you say,' said Boromir. I have seen a bright ring in the Halfling's hand; but Isildur perished ere this age of the world began, they say. How do the Wise know that this ring is his? And how has it passed down the years, until it is brought hither by so strange a messenger?'. FotR:The Council of Elrond
As @MattGuring his pointed out, it then takes months for him to get to the point that he's willing to break his oaths as a loyal warrior of Gondor.
There were only two Men at the Council of Elrond - Aragorn and Boromir.
The full tally of attendees is given at the start of the chapter and may be quoted here:
He then pointed out and named those whom Frodo had not met before. There was a younger dwarf at Glóin's side: his son Gimli. Beside Glorfindel there were several other counsellors of Elrond's household, of whom Erestor was the chief; and with him was Galdor, an Elf from the Grey Havens who had come on an errand from Círdan the Shipwright. There was also a strange Elf clad in green and brown, Legolas, a messenger from his father, Thranduil, the King of the Elves of Northern Mirkwood. And seated a little apart was a tall man with a fair and noble face, dark-haired and grey-eyed, proud and stern of glance.
There were therefore no other Men to be tempted.
For starters, we aren't privy to the thoughts and inner dialogues of the men around the table, so it's not fair to assume they aren't tempted. For all we know, several of them could be thinking about how valuable of an asset the ring would be to their prospective kingdoms/represented groups. Boromir, in fact, says as much... with only Aragorn showing any disagreement.
Also, with regards to the movie version, the un-named older men sitting near Boromir are said to be tradesmen from Dale (Laketown) and other settlements. Many of them may not know of the ring or its power, beyond what is whispered in legend. Even so, Sméagol seems to be the only individual in the entire epic who is affected by the ring instantly. All others appear to succumb to its power gradually. Others must be exposed over time.
On a side note, since reading your question I now have this insane image from the movie of one of the old guys snatching the ring off the tree trunk and running off shouting "Mine!".
1.) Temptation over time
The ring works over time. For example, as given in another answer Boromir changes only after a significant period of time. The same can be said for Smeagol who slowly transforms into Gollum over a VERY long period of time. Bilbo while tempted by the power of the ring doesn't suffer any negative effects for months and is able to hold the ring for years.
...is not a normal place. This is one of the last great strongholds. It may look like a peaceful village but it is guarded by Elrond's magic and those of his closest councilors and has managed to keep out all sorts of unsavory sorts for several hundred years. It is a magic fortress and we would expect that the enchantments that protect it also work to abate the effects of the ring.
Those present at the Council of Elrond were leadership reps for their respective branches of people. They were all agents for good, and all were there because of Sauron's attempts to get in their business. Elrond is speaking about how they all came together with no invitation:
You will learn that your trouble is but part of the trouble of all the western world. The Ring! What shall we do with the Ring, the least of rings, the trifle that Sauron fancies?
That is the doom that we must deem. That is the purpose for which you are called hither. Called, I say. Though I have not called you to me, strangers from distant lands. You have come and are here met, in this very nick of time, by chance as it may seem. Yet it is not so. Believe rather that it is so ordered that we, who sit here, and none others, must now find counsel for the peril of the world.
These people, excepting maybe Boromir and his huge ego thinking he could master anything (but already at the Council he is being mastered by the One's influence) knew this to be something abhorrently evil. The cause of so much of each of their people's problems