That seems highly unlikely. Stalin was born and raised in Gori, a city which is some 80km west of the capital Tbilisi, and today about an hour's drive. While it isn't exactly a large city (it has around 50,000 people today, and would have been considerably smaller in the late 19th century), it isn't some remote village where time stopped. And to add to that, Georgia as a whole may have remote regions and relatively poor, but they aren't like that. I suspect the idea from this came from a book written by Richard Halliburton in 1935, Seven League Boots. I'll quote here from a non-historic site, but it is an accurate quote:

On page 162, in the chapter ‘The Last of the Crusaders’ Halliburton wrote:

"In the spring of 1915, some months after Russia’s declaration of war against Turkey, a band of twelfth-century Crusaders, covered from head to foot in rusty chain armour and carrying shields and broad-swords came riding on horseback down the main avenue of Tiflis [Tbilisi]. People’s eyes almost popped out of their heads. Obviously this was no cinema company going on location. These were Crusaders – or their ghosts.

The incredible troop clanked up to the governor’s palace. 'Where’s the war?' They asked. 'We hear there’s a war'.

They had heard in April 1915 that there was a war. It had been declared in September 1914. The news took seven months to reach the last of the Crusaders…"

The site goes on to explain they were Khevsuretians, people from the region of Khevsureti, in the north-east of Georgia. It also notes that this is highly unlikely, as no one else described Georgian crusaders coming to Tbilisi in 1915 to fight the war. However the myth persists, in part because there is a photo of them dressed up in Crusader attire (see the link), though I don't know the context of that. It's also worth noting that rather than wear some chain-mail, it would be more likely for them, and anyone Stalin may have seen, to wear the chokha, the traditional clothing of the Caucasus, which is notable for its pockets along the chest to hold cartridges. That very well could have been something young Stalin saw growing up, and he certainly would have at more formal events (weddings, funerals, and the like), as it is still done today, but I seriously have doubts on chain mail being worn by some locals walking around Gori. It seems more like something to spice up a book aimed at a younger audience (as this book implies, based on the description provided by Google), and based on a questionable report by an American from the 1930s.

For medieval Europe: a striking feature of medieval romance literature (starring, frequently, knights) is that as soon as travelling knights arrive at their destination, they take off their armor. The texts mention this explicitly, a rare appearance of this sort of detail, but don't make a big deal out of the process. Just, Gawain arrives and takes off his armor and goes to see the king. (I'm thinking here of episodes in the early 13th century Vulgate/Lancelot-Grail cycle of texts, which predates the gradual adoption of plate armor). This would seem to suggest that mail was not a thing to be worn indoors.

I can't say how typical it was to wear mail while traveling. The knights of the Arthurian saga are always running into trouble while abroad even in their homeland, and the evidence from Crusade chronicles is somewhat mixed.