There are a lot of isolated towns and cities in the northern reaches of European Russia, Siberia and the Russian Far East. Many of those town started out as Gulag camps and turned into mining towns.
The most isolated city in the European part of Russia is Vorkuta, which is also the easternmost city in Europe (same longitude as Afghanistan). Some maps show a road going up from Syktyvkar, but apparently it is not really motorable most of the year.
The isolated city of Salekhard on the Ob River, is now accessible by road, as they built a road network in the Yamalo-Nenets Autonomous Okrug to support the oil & gas industry. Most people probably have never heard of Yamalo-Nenets Autonomous Okrug, but it is an area larger than Texas. There are a surprising number of sizable settlements in this remote region.
Further east, there are mining or forestry towns like Norilsk and Igarka that are not accessible by road. Igarka has seen a population collapse because its main industry closed. Norilsk has made headlines for the extreme ground and air pollution in the region.
The city of Yakutsk is not accessible by road, as there is no bridge over the mightly Lena River. However a new road is being developed from the west, called the 'Vilyuy' or A331.
As Duke87 mentioned, there is also Petropavlovsk-Kamchatsky, which is probably the most isolated out of all large Russian cities, since it's very far from the continental road or rail network. Then there are towns like Anadyr, which is the capital of Chukotka, which is the easternmost administrative division of Russia. The nearest continental road is the road to Magadan, over 1,300 kilometers away.