In Iceland, 1000 AD will most likely be remembered as the date of Christianization. This took place over many decades, of course, starting with missionaries in the 800s, but the Alþing (the legislature) made the decision to have Iceland be a Christian state in 999 or 1000.

Would the average islander have been Christian in 1000? Archaeologist Orri Vésteinsson tells the account of a priest, Þormóðr, who was the sole priest on the island in 1000 and was tasked with baptizing the whole population. Vésteinsson says "it must have taken a long time to establish any basic Christian practices," which speaks not only to the prevalence of Christianity (likely low), but also the sparse nature of settlement. He goes on to say that there are only five churches on the island recorded as being built before 1100, missionary bishops not arriving until 1020, and so on. Vésteinsson caveats all of this information by saying the empirics are few and far between to give us a clear picture. He's making the best with what he has at his disposal, as is so frequent in cases like this where the record is thin and the supporting material is practically nonexistent.

Even though our knowledge of the religious state of the country may be shaky at best, it may be the part of millennial Icelandic life we know the best. The Íslendingabók, a national history written in the 12th century, tells of major milestones in the country's history. Christianization is part of this, but the daily life of people 100+ years prior to the writing of the book, which itself is a rather slim volume, doesn't make the cut.

So what was life like in Iceland around 1000 AD? The country was in the process of Christianization, which was about to be accelerated by decree of the legislature. Daily life, however, we don't know much about (or at least I can't remember reading other sources that have spoken about it).

Sources:

Karlsson, Gunnar. The History of Iceland. Minneapolis, MN: University of Minnesota Press, 2000.

Vésteinsson, Orri. The Christianization of Iceland: Priests, Power, and Social Change, 1000-1300. Oxford: Oxford University Press, 2000.

If you're interested, you can also check out a (maybe not quite perfectly) translated version of the Íslendingabók here: https://en.wikisource.org/wiki/%C3%8Dslendingab%C3%B3k

I'll chime in as this has been a large obsession of mine for a few years now. While the more highly-upvoted stories are very interesting and entertaining, they are somewhat inaccurate in terms of details and a bit more saga-like in their style.

If you're interested, the best English-language source that I've found (that's still readable in a somewhat entertaining and interesting, non-purely academic way) is Wasteland With Words by Sigurður Gylfi Magnusson.

In short; life is pretty miserable. For most residents, filth, disease, and health issues are common due to the living conditions: small, cramped, dirty, cold. Turf-roofed huts that leak, constant wind and cold, fast spread of disease due to close quarters. It was a very very hard life for most non-wealthy folk.

Iceland had a very interesting societal structure involving who owns what; there are bailiffs who are charged with keeping their local districts in check, and in many cases if a husband died, that farm and the wife would become the property of the bailiff.

For farmers and fishers, death and injury was fairly common. Some of the most interesting parts of life were merely staying alive. Fishing back then was utterly insane compared to today; small rowboats and sheepskin or cowhide 'wetsuits' that provided little or no protection.

Diets were very bland. Butter was often so highly prized that it was used as currency, and sugar was an exceptionally rare treat. Dried fish and (somewhat rarer) meats were the staple throughout winter. Much of the food was rotten.

It's not a place you'd want to live in! Another very interesting read is Halldor Laxness' Independent People which is widely considered to be one of the best stories of Icelandic life before modernization. All of what I've mentioned above can be sourced from Wasteland With Words, which I really recommend that you pick up if this is something that interests you. A trip to Iceland would be in short order after!

As a further question, how many people would have lived in Iceland back then? As it is very sparsly populated today, I imagine the population must have been extremely small back then?