This isn't that improbable, indeed, in some ways it is a bit of an unlikely modern miracle that Russia stayed as in-tact as it has thus far. The people in-charge were either an alcoholic career Communist apparatchik who jumped on the nationalist bandwagon for power and was past his prime (Yeltsin), number-crunchers who didn't really know what they were doing (Gaidar), wannabe strong-men (Lebed , Zhirinovsky) or shady KGB/intelligence types in the background (Viktor Yerin). The fact this group of people kept Russia alive despite every kind of disadvantage is a major achievement. It is a lasting tragedy that we now find ourselves in the position we do, waiting for Putin to either leave office or die in-order to allow the Russian people the opportunity to take control of their own country once again.
Any leader, whether Yeltsin or anybody else, would fight any change tooth and nail. Whatever the man's obvious weaknesses, he was not afraid of a fight, he also had some kind of weird moments of competency on occasion, for example out-playing Gorbi and enough connection with the ordinary people when it really mattered, particularly during the death of the USSR, the 1993 constitutional crisis and to a lesser extent the 1996 Presidential election and his resignation at New Year. He showed himself willing to use military options to suppress revolt within the country. Let's assume that the PoD is something happens to Yeltsin. I don't see a natural successor. There's a lot of men who'd like to be, but Yeltsin still had credibility in 1991, he was the man who ended the USSR, his reputation was still in-tact, he came out of the confrontation with the hardliners in 1991 and his dramatic standing on the tank, a greatly enhanced, global figure. He looked the part. He still had that benevolent drunken uncle thing and just the hint of what he might have been had he been sober (and by all accounts he was definitely an able man in his youth). Something happens to Yeltsin during that period, he is injured seriously enough to sideline him or killed, EVERYTHING changes.
There would inevitably be a power vacuum amongst the anti-Communist side as they struggled to find somebody who could invoke the right kind of support amongst the people. They would be left with apparatchiks and shady types who would be perceived as more of the same didn't have the charisma that Yeltsin was blessed with when he was together and cursed by the memory of what he might have been when he was together. None of them would have a brand and would have to rely on underhanded tactics and the kind of backhanded, KGB-style crap that got Russia into this mess in the first place. Yeltsin would become to democracy in Russia, what Lenin was to Communism there. His successor would either become another Stalin or would stage a couple and be done with it. It's perfectly possible there could be competing alternate groups, all claiming to be the legitimate government of Russia. Who would the international community deal with? Yeltsin survived partly due to his past actions between 1987-1991 and his demonstrated willingness to use force in 1993 and in 1994 in Chechnya. He also survived because the US and China especially were prepared to continue to deal with him and his team, at least in-part because there was no other viable alternative they could stomach. Take Yeltsin out of the picture and this changes. The pretence of stability even if it is just a pretence has gone which means there is much less willingness to take risks, meaning the investment Russia desperately needs is not simply there and we have squabbling groups of apparatchiks and Generals squabbling over scraps.
No matter what the case, 1) democracy is done for, 2) the gloves were probably totally off and 3) the minorities of Russia don't have anything to gain anymore. Legitimacy has to come from somewhere else now, probably a heady-brew of nationalism and Orthodox Christianity, right away excluding significant populations in the periphery now, with lasting implications in places like Chechnya, Ingushetia, Dagestan, Karelia and others, that might dream of independence, to organise and gather support.
Karelia is right next to Finland and it's population has much more culturally in-common with Finland than they do with ethnic Russians. Their language is mutually intelligible, some of the border areas in the Karelian isthmus for example, were taken from Finland during the Second World War and reaches very close to St. Petersburg indeed. A system which is all about the Russian mother-land and Russian identity is not going to appeal to these people. Yes, integrating such a large area and population into Finland (about 172,400 km2 and 643,548 per the 2010 census) is a HUGE and expensive task, but they would likely see it as an appropriate response to what was happening. From a geopolitical perspective, it is also deeply significant as eastern Karelia basically cuts Russia off from Murmansk, its strategic Arctic port. A serious attempt by Karelia to join Finland (which would not surprise me and why wouldn't they given that the Finnish standard of living is higher?), not only complicates relations between Russia and Finland, but would inevitably engage the Scandinavian states and the nearby Baltic states as well. Finland has not yet joined the EU an remains outside NATO to this day. In this universe, resources are expended by the democratic powers to support Finland based on perceived threats of a resurgent Russia and couched in-terms of national self-determination for audiences at home, speeding up Finnish accession to the EU in 1994 and possibly making a strong case for NATO membership in the same way that the Baltic states were so eager to join, leaving Sweden the sole Nordic non-member.
At the same time, there's no reason the chaos at the top doesn't buy time for the Tartar/Muslim areas where there was a large enough proportion of non-ethnic Russians to make this feasible. Perhaps Muslim powers getting involved and/or fighters from Afghanistan coming over. The Russians made a rather underwhelming showing in Chechnya IRL, so perhaps if the Chechens had been a little more lucky/gotten some more support from a Sunni Islamic power looking to assert itself, maybe the Saudis or somebody nuts like Gaddaffi decided to funnel oil money their way to leverage power.
A more successful Chechen revolt between 1991-1993 would have made a world of difference and make other peoples start to believe things were possible. This would inevitably have consequences. The first of which is far more political and economic instability. Russia is already incredibly difficult to govern at this point, the economy is in free-fall, they have no set sense of the direction they want to go in, people's savings have been wiped out and they aren't getting paid, the military and police will likely be restive, if not outright corrupt, strong-men might emerge at any given moment. Add in that Russia would need FAR more foreign attention just to keep things like the nuclear arsenal, submarines and the space program functional until an alternative can be instituted, which means the major democratic powers of the US, the UK and France who were active IRL, being MORE active. This is obviously going to be a mess. Aside from the fact IRL, they are relying on the alcoholic Yeltsin and his cronies, to keep it together (it's a miracle the man did as well as he did frankly), there's a good chance that extremist nationalism and alternate socialist philosophies make come-backs here. The Communists might be positioned to do really well in any prospective Duma election. The chances of a coup d'etat led by some disaffected Generals of the Right or the Left is really quite high in this vacuum. In that sense Russia would resemble an African nation or somewhere in South America in its governance more than a European one.
Certainly there would be considerably less loyalty towards the government and Moscow across the board. Disloyalty amongst minorities could probably be assumed, particularly if people like say the Karelians decided independence (or in their case probably union with Finland) was in their best interests. At the very least now is a very good moment to leverage concessions from Moscow and start actively decentralising the Russian state. Governance in the Caucuses might actually breakdown altogether and there might well be a real risk of regional wars there if the Armenian-Azerbaijani mess is any indication. Certainly the military is going to struggle to retain its personnel as they cannot pay their soldiers and now must concern themselves with the loyalty of minorities. Additionally, organised crime will likely target personnel for recruitment as security, drivers, couriers and such, an easy if incredibly immoral pay-day many men, some of them already nursing the beginnings of the national curse that is alcoholism, some desperate to get out of the military and some just desperate for an income, will likely take up the offer, especially given the economic meltdown that seems inevitable. Given the Armed Forces' well documented problems with dedovshchina, it's difficult to see how the transitioning Russian Armed Forces with really serious security problems and probably taking quite an overt political role as well, are equipped to address this.
Other than shared ethnic ties, the peoples of Siberia don't necessarily have that much in-common with those of European Russia. They have other priorities, so its possible that more success amongst minority breakaway groups might encourage regional leaders in places like Irkutsk, Krasnoyarsk, Omsk, Vladivostok and others look at their position and conclude that Moscow needs them more than they need Moscow. Given the vast untapped natural resources there and the willingness of Western firms to exploit those oil, natural gas and precious metal reserves, they would be in a really strong position to negotiate at the very least a high degree of federalism from Moscow, however in the context of a weaker Russia, they might decide to cut their losses, face towards the Pacific and make nice with the Americans, Japanese and South Koreans.
Other problems probably include, tracing and protecting Russia's enormous NRBC stockpile, containing an explosion in Russian organised crime that rose up to fill in the vacuum, deciding who to actually make deals with out of all the conflicting power-players and potentially disputing Provisional Governments and dealing with the implications of a worst Russian great recession, probably a full-blown depression on the world's financial markets, which would be devastating to savers and pensioners across the world.
Basically it's a hot, bloody mess.