Possibly people in Irkutsk do not need much EURO?
It's of course possible to exchange currency in Irkutsk. It's a big city with a population of some 600 thousand, roughly the size of Helsinki, Stuttgart or Lisbon, so it certainly has a lot of banks and currency exchange points, any of which can be expected to accept Euros. A brief search shows 105 exchange points in the city, you will surely be able to find one close to your location in the city, especially if you arrive by rail, which is a major transportation method there.
This is a page showing the current exchange rates for EUR in Irkutsk. The three major numbers, left-to-right, are: the central bank's exchange rate, best buy rate in the city, best sell rate in the city. As of this moment, the best rate for selling EUR is at Гринкомбанк (Grinkombank), at 74.1 rubles per Euro, with the official exchange rate of the Russian central bank being 73.63.
You will not trade with people, you'll deal with banks. Most of them, probably all, will provide currency exchange service.
However, in the recent years more and more shops, hotels and restaurants began to accept international bank cards (Master Card and Visa). I was surprised when even a small shop in a tiny village on Baikal coast allowed to pay by card. Taking this into account, you may not need to exchange a large sum of money: currency exchange will be performed transparently.