See subject. The Cold Stare, the Icy Glance. I know we’re mostly water.
It’s not just the ambient temperature; it’s also wind-chill, which is highly relevant to frostbite, since as you say, we’re mostly water.
I don’t know if there is a specific number, but typically here in Canada, when the temperature and wind-chill drop below a certain point, the helpful little gnomes on CBC weather add, often in a gleeful tone of one safely ensconced in a nice warm broadcast booth, something like “And remember folks, at this wind chill, exposed flesh will freeze in 1 minute!”
I’ve never heard of eyes freezing, since the eyelids’ blink reflex kicks in in bad weather, particularly cold wind. But if you keep your eyes closed too long, the tears might start freezing up, as the narrator of Sam McGee commented:
“If our eyes we closed,
Then the lashes froze,
Till sometimes we couldn’t see.”
Frostbite on the cheeks and fingers can happen very quickly, but hoods, scarves, balaclavas and good mittens/gloves are pretty effective. That “exposed flesh freezes” line usually means just that - completely exposed.
I’ve had frostbite on my cheek just from the wind and low temperatures.
I’m about to go out into -19 C to get the snowblower going; haven’t heard if there’s any wind chill, but it doesn’t look like there’s much wind.
Here’s the NOAA chart. Imperial, not metric.
What it does not take into consideration is circulation/insulation below the skin. For example, the skin over cheekbones tends to freeze up more quickly than the skin further down on the cheeks.
One of my skiing goals is to make it to -100 F combined temp and wind-chill. Been very close, but not quite there yet.
Oh, and the other thing that the chart does not take into consideration is variation of sunlight.
Agreed with Muffin. Sunlight has a huge effect.
It can be -10 Celsius, and my wooden deck surface is +50 in the afternoon sun, dry as a bone. You can stand out on it in a T-shirt.
I knew it was warm, but I didn’t realize how warm till I got a laser thermometer. Pointing that at the wood gave me those temperatures. Caveat: it is quite dry where I live.
Skin and tissue freeze at slightly below 32F or 0C, just like anything else comprised primarily of water. Getting to that temperature involves a lot of different factors.
That is indeed a very subjective question. Are you going for a walk in a cold temperature? Walking with the wind, you’ll probably be quite comfortable in your parka and headband. Walking against the wind, the hood comes up and your scarf goes over your face to prevent frostbite.
How fast are you walking? How long have you been walking? Once your body starts warming up from exercise, your heart is pumping faster and your skin gets warmer and you might even shed a layer of covering. If you’re just standing and waiting for a bus, you’ll probably want all exposed skin covered.
A funny thing that people who aren’t used to the cold might not know about; when you’ve been out in the cold for a while, your face muscles slow down. You smile, and it takes a bit for your smile to come back down again. 
And I always thought I was imagining that.
Shhhhh! Mum’s the word on that. We want them to keep on believing that Canadians are friendlier.
If you ask someone coming in from outside what the weather it like, you are entering frostbite territory when they reply “It’s cold as fuck,” as opposed to “It’s cold.” There’s a lot to be said for observing what others are doing and how they are dressed, and then copying them.
You’re quite mad - you know that, don’t you? 
I heard some under ice swimmer’s eyes froze when he was trying for a record. Has anyone heard of that?
I know it’s less than 10F when you walk outside and your boogers freeze.
I doubt that an eyeball would freeze while a person is swimming. The front part of the eyeball is exposed to air that is far colder than liquid water without it freezing. Most of the eyeball is nicely protected and insulated inside the head. Just a few data points, but I have fished out quite a few people from ice-out water, and swum myself in those condiditons, and never came across a frozen eyeball. My best guess is that a person would seize up from hypothermia and drown long before an eyeball froze.
Jeez. Now that’s empirical evidence.
Let’s look at it another way. An eyeball in liquid water would not be colder than liquid water, which would not be frozen, so the eyeball would not be frozen.
Ocean water can be colder than 0C or 32F without freezing because of the salt content. I can’t see a whole eyeball freezing, but maybe someone swimming in freezing cold ocean waters had their eyes freeze shut. Or perhaps they were talking about some other kind of balls.
I expect that eyeballs have a greater salt content than seawater, but just a guess.
I have heard of eyeballs freezing; apparently it’s something polar workers and himalayists are used to taking into account and one of the reasons to keep your googles on (newbies tend to take them off because they mist up at first). It’s pretty weird, hearing someone tell how they were climbing and, when they complained about the mist, their cordmates said “but… it’s sunny! Oh SHIT, your eyeballs are freezing! We’re going back now.” Even weirder when the next sentence is “the other times my eyeballs started freezing I was able to recognize the symptoms right away and stave it off.”