So I have overheard a conversation where some guy wondered if it's possible to swim in Arctic Ocean, the only ocean he didn't do yet.
I actually have an answer, but maybe there are better options. Are there any accessible beaches on any of Arctic ocean seas where you can reliably swim? I expect the location to be accessible by car or, preferably, also by public transport, and warm enough to function as a beach resort at least eventually.
Scandinavia is brushed by the Gulf Stream and therefore warmer than other regions at the same latitude. So the north of Norway is a good candidate. Norway is part of the Schengen Area.
The south-west of Norway lies on the North Sea which is considered part of the Atlantic Ocean. Most of the Norwegian coast is on the Norwegian Sea which is considered part of the Arctic Ocean. The north of Norway lies north of the Arctic Circle.
The Lofoten Islands are a fairly popular tourist spot which lies a little way north of the Arctic Circle. It has beautiful beaches and other scenery. You wouldn't want to go swimming without a wetsuit, but you wouldn't be the only one taking a short dip in summer.
The water temperature in summer is never very warm, but in summer, you shouldn't risk hypothermia without warning signs. The warm period is roughly mid-July to late August, with water at about 11–15°C (and air about the same temperature!).
You may even enjoy a dip under the midnight sun until 18 July.
My own answer as follows: The beach of Ягры, Северодвинск, Russia allow you to swim in Arctic Ocean's White Sea while overlooking occassional nuclear submarines passing by.
I have been there and it's very possible to swim, even if waters are not very warm. However, there are at least two downsides to this answer: the town is problematic to visit without Russian citizenship, and it's located on very closed off White Sea which may be argued to not be "Arctic Ocean proper".
Otherwise, accessible by car, by bus from Archangel or by train plus some local bus. OSM of Yagry beach.
I have gone swimming at the beach in Churchill, Manitoba.
I am not sure how reliable the swimming is. I was there at the end of August, which was so off-season that the main coffee shop in town had closed for the week. The proprietors of my bed and breakfast thought I was crazy, and accompanied me for safety. I remember the water being cold, but not as cold as some places I've swum in the Adirondacks.
Regarding safety, the beach is directly behind the hospital, but it is also covered in signs warning of the dangers of polar bears. A day or two later, a few miles away, a man was mauled by a polar bear while he was on a ladder against a building.
It's funny you say "preferably by public transit", because you can't get there by car. There are roads in Churchill, but they don't go far enough to get to any other towns. The road for which you do not need to take special precautions ends in Thompson, and routes for cars of any kind end in Gillam, but I was able to take a train from Churchill to Winnipeg. That train took 2-3 days, but does connect you to more normal transit options.
Churchill is definitely a tourist destination, but for people wanting to see polar bears, beluga whales, the northern lights, and birds. I spent some of my time there in a t-shirt, but I think most people would not consider it "warm enough" to be a beach resort. I remember meeting an Italian lady there, who gave the impression that she had not yet thawed from the preceding winter. Similarly, I went swimming at Deer Lake, which has a beach without the polar bear warning signs. I remember others using it as a beach, but I don't remember anyone else going in the water.
