Sauron is a Maia, sort of the Middle Earth equivalent of an angel (or fallen angel, in this case)1. So the real question is "can an angel be killed?" The closest we have to a comparison are the balrogs.
Balrogs are usually considered to be Maiar, fallen angels corrupted by Sauron's old boss Morgoth long before the events of Fellowship. And balrogs can be killed, but it's not easy. I can think of three instances where the death of a balrog is explicitly stated:
- In The Silmarillion Ecthelion, an Elf-lord of Gondolin, kills Gothmog, captain of the Balrogs
- Also in The Silmarillion, Glorfindel of Gondolin kills another balrog
- In Fellowship, Gandalf kills the balrog in Moria
These events have two things in common: all of the beings who killed a balrog had once been in Valinor, Middle Earth's heaven, and they all died in the attempt. So it seems as though killing a balrog is possible, but exceptionally difficult; it takes a special kind of person.
By the time of the Last Alliance, I think the only remaining elf who had dwelt in Valinor was Galadriel2 (The Wizards hadn't yet been sent to Middle Earth at the time), and she wasn't going to be fighting in the battle.
So no individual at the battle was capable of killing a lesser Maia, much less Sauron, a very powerful Maia, short of some astonishingly good luck. So could he have been overwhelmed by the entire army?
Sauron only lost one other conflict in the Second Age3, mostly because of the great Númenórean host at the height of their power and influence:
[I]n the very nick of time the great armament of Tar-Minastir came in; and Sauron's host was heavily defeated and driven back. The Númenórean admiral Ciryatur send part of his ships to make a landing further to the south.
Sauron was driven away south-east after great slaughter at Sarn Ford (thr crossing of the Baranduin); and though strengthened by his force at Tharbad he suddenly found a host of the Númenóreans again in his rear, for Ciryatur had put a strong force ashore at the mouth of the Gwathló (Greyflood), 'where there was a small Númenórean harbour'. In the Battle of the Gwathló Sauron was routed utterly and he himself only narrowly escaped. [...] [B]roken and humiliated he returned to Mordor, and vowed vengeance upon Númenor.
Unfinished Tales Part 2 Chapter IV "History of Galadriel and Celeborn"
It also probably didn't hurt the Númenórean cause that the One Ring was still very new to Sauron, and he presumably had not quite mastered it.
However, centuries after this event the Númenóreans fell, and lost the blessing of the Valar. By the time of the Last Alliance, Elendil and Isildur are the closest things to true Númenóreans still alive, and it takes both of them to take them down, with nearly-fatal effort. In Fellowship, Elrond (One of the few participants in that battle alive at the time) recounts the story of Sauron's fall:
Gil-Galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword.
There are lots of ways to interpret that account, and Elrond is clearly condensing the story for the sake of convenience, but three things are clear:
- Elendil died battling Sauron
- Sauron was not completely defeated by Elendil4
- Isildur had to cut the ring from Sauron's finger to end the battle
It took two men of Númenor, both of whom were alive when Númenor sank, to subdue Sauron. Their descendants at the time of Lord of the Rings have much-diluted Númenórean blood - I'd say that Aragorn is the only living man who can come even close to matching Isildur or Elendil and, as much as Sauron fears Aragorn, it's doubtful that he alone could match his ancestor's combined feat.
However, it is possible for Sauron to lose his form (about as close to death as a full Maia can get). This happened at least once, during the Second Age. Sauron was on the island of Númenor, and convinced the Númenóreans to invade Valinor. They failed, obviously, and Ilúvatar himself sunk Númenor in punishment. Sauron was stuck on the island at the time, and although he wasn't utterly destroyed he lost the ability to assume a pleasant-looking form.
The tl;dr of all this is that it's not inconceivable that Sauron could be forced out of his current form by a sufficiently large and powerful force. However, it's unlikely that any such force remained in Middle Earth by the time of the Last Alliance.
However, even if they succeeded Sauron would not be dead; it's important to note that even after the destruction of the Ring Sauron didn't die: in Return of the King, Gandalf says that:
he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but can never again grow or take shape
1 If we're drawing parallels to Judeo-Christian cosmology, it would be better to think of Maiar as a lesser order of angels; strictly speaking the Valar, often considered the "gods" of Middle-Earth, are more accurately classed as angels (With Eru Ilúvatar as The One True God), and Valar and Maiar are technically the same kind of creature, just with a different level of power.
2 As Mike Scott reminds me in comments, there's some debate over whether there are two Glorfindels or just one. See my comment here for the background knowledge. So it's possible that there was another elf walking around with the ability to at least kill a lesser Maia. Whether or not he could defeat one or Sauron's strength is purely speculative, because we no longer have any canon support.
3 Sauron did, of course, lose occasionally in the First Age, but very infrequently and always against some very powerful beings. The most famous time is probably in the story of Beren and Lúthien, when Sauron is pressured into yielding Tor-in-Gaurhoth to Lúthien. In this instance he's subdued by Huan, a wolf-hound who formerly belonged to the Vala Oromë, and had been granted special powers. Not an ordinary fight.
4 You can argue over how much Sauron was weakened by his fight with Elendil, and the Jacksonverse argues that it wasn't very much - he had enough fight in him to curb-stomp Isildur for a while - but it's clear that Sauron was still alive and kicking when Elendil was killed. What's not clear is how much effort it took Isildur to cut off the ring; it could have been little, it could have been nearly fatal.