This question was discussed by R. Gabriel in one of his books (I referred to them in this thread:
http://slinging.org/forum/YaBB.pl?num=1257730993/0Gabriel's point was in fact that the lethality of slings and arrows is not that high. He gets that slingstones and arrows are high lethality weapons, but notes that in no ancient battle was the main line of heavies mowed down. So Gabriel's assumption is that missile effectiveness is reduced by armour and shields-- probably considerably so. As confirmation of this, we might turn to Xenophon, who in his Anabasis provides an eyewitness account of what it's like to fight against slingers:
[3.3.6] After this they took breakfast, crossed the Zapatas1 river, and set out on the march in the formation decided upon,2 with the baggage animals and the camp followers in the middle of the square. They had not proceeded far when Mithradates appeared again, accompanied by about two hundred horsemen and by bowmen and slingers--exceedingly active and nimble troops--to the number of four hundred. [3.3.7] He approached the Greeks as if he were a friend, but when his party had got close at hand, on a sudden some of them, horse and foot alike, began shooting with their bows and others with slings, and they inflicted wounds.
-- wounds, and not fatal casualties. I also would mention Thucydides 2.81.8, describing a campaign in W. Greece in 429:
But when the barbarians in their flight broke in upon them, they took them in and uniting their two divisions kept quiet there during the day, the Stratians not coming to close quarters with them, because the rest of the Acarnanians had not yet come to their support, but using their slings against them from a distance and distressing them; for it as not possible for them to stir without armour; and indeed the Acarnanians are famous for their excellence in the use of the sling.
The assumption seems to be that if you wear armour, you're actually somewhat safe from sling sniping. My own view is that the battle winning, ground-occupying, assault infantry which closes with the enemy and kills him or drives him off are, by nature, pretty heavily armoured (this also comes from reading stuff in another forum, fioredeiliberi.org); light infantry often runs off. (Caesar doesn't, I think, describe naked chaps in battle; that's Polybios in book 2, about Telamon. Caesar's opponents are pretty heavily armoured, if the archaeology is indicative).
So: body armour (of any type: plate, or flexible— mail, linen, padded), helmet, and shield, are sufficient to greatly neutralize missiles of any type (slingstones, hand thrown stones, arrows); as Morningstar often observes on this forum, the real killer is the javelin, which has enough mass and punch to give even the armoured chaps problems. I'd add that the other way of neutralizing missile troops is another line of missile troops, which produce the characteristic pre-battle skirmishing between lights-- described by Thucydides in Sicily, or by Polybios at Cannae-- inconclusive.