The following isn't an answer telling us what Russian public opinion wrt to the "special military operation" is. It is listing some approaches that have been tried and info that has come out. As noted, there is a lot of "can poll responders really be trusted to speak their mind?" concerning supposedly high support for the "special military operation". Lots of people are asking themselves this question
*, the actual answer seems hard to know.
One possibility, but it is very open to manipulation, is to look at leaks of polls supposedly commissioned by the Russian government itself, for internal-only use. Presumably, if only to inform their decision making, the Kremlin ought to minimize the poll's bias.
Meduza.io published an article about one such poll, supposedly conducted by Russia's Federal Protective Service (FSO), for internal governmental consumption only, allegedly saw support for peace negotiations go from 32% in September to 55% in November ( Meduza.io @ mediabiasfactcheck.com : left bias/high factual reporting ) .
That would coincide with a string of recent un-hideable losses and the start of the limited mobilization, so it is not totally implausible.
The flip side is that it will be extremely hard to confirm that this was indeed a real poll and that the results were really those mentioned. As opposed to a manipulation by Meduza.io which has a large pack of dogs in this particular fight.
In the same vein, in the past, there had been a report of the actual referendum result in the 2014 Crimea annexation and it was wildly at odds with the official numbers.
the official Crimean election results, as reported widely in the Western press, showed a 97 percent vote in favor of annexation with a turnout of 83 percent.
the website of the President of Russia’s Council on Civil Society and Human Rights (shortened to President’s Human Rights Council) posted a report that was quickly taken down as if it were toxic radioactive waste. According to this purported report about the March referendum to annex Crimea, the turnout of Crimean voters was only 30 percent. And of these, only half voted for the referendum–meaning only 15 percent of Crimean citizens voted for annexation.
Since September, you also need to look at public opinion differently in this case then you might for Western countries during the Iraq wars or Afghanistan. In those wars, those doing the dying were professional soldiers, who had signed up for it. Asides from their friends and family, the general public, if not holding strong pacifist/anti-imperialist opinions might be concerned but it did not have much day to day impact on them. To some extent this was the "special military operation" before the mobilization.
Since September, a much broader slice of the Russian population has "skin in the game", for themselves or their acquaintances. Rather than Iraq/Afghanistan an - imperfect - lens to evaluate Russian opinion from a Western PoV might be more Vietnam, where there was a draft. And one common data point might be the number of draft dodgers who fled abroad.
Since I found another approach while responding to comments, I'll share it. Note that the source, rferl.org, isn't - quite - going to be unbiased either. I mention it for completeness but have no real opinion on reliable an indicator it would be. This dates from April btw, when things were going much better for Russia:
In their experiment, the researchers used an online Russian-designed sociological tool called Toloka to recruit 3,000 adults and devised a list of questions asking respondents whether they supported one or more of four social policies: same-sex marriage, abortion restrictions, the war in Ukraine, and cash welfare payments for poor Russians.
Respondents aren’t asked to say which policies they support, merely how many of the four items they support.
In this survey, which was conducted on April 4 and which sociologists broadly call a “list experiment,” half of the respondents were given a three-item list, with the question of the Ukraine war omitted; the other half was given a four-item list that included the Ukraine war question.
The researchers also asked respondents a straightforward, yes-or-no question: “Do you support the war?”
The results showed that when Russians were directly asked the question “Do you support the war?” 68 percent said they did. When using the list experiment, however, support for the war dropped to 53 percent.
“Do Russians tell the full truth when asked about their support for the war?” the researchers wrote. “Based on our experiment, we can safely conclude that they do not.
About the impossibility of conducting such a poll, "because special reasons about the nature of asking support for/disapproval against war": YouGov's tracking of polling support for the Iraq War:
Though it has been controversial for over a decade, the invasion was actually popular at the time. In 2003, YouGov conducted 21 polls from March to December asking British people whether they thought the decision by the US and the UK to go to war was right or wrong, and on average 54% said it was right.
But more than 10 years of opposition is a long time, and many people now remember things differently. Now only 37% of the public say they believed military action against Saddam Hussein was right at the time, instead of the 54% recorded at the time.
And details about the poll timelines. The question itself? Nothing very complicated: Do you think the United States and Britain were/are right or wrong to take military actions against Iraq?
See, no rocket science, these questions can be asked, when you have an open enough government.
There is nothing inherently unanswerable about this question. What makes it in unanswerable is the nature of Putin's regime.
Update 230203: Podcast from Meduza - Beyond TV and polling in Russia is a good listen.
Dr. Morris, a professor of Russian and Global Studies in the School of Culture and Society at Aarhus University in Denmark, argues that researchers should devote more attention to less controlled platforms on social media and exercise more caution when generalizing based on survey data collected in Russia.
Dr. Morris, specifically casts some doubts about Levada.
* Lest one underestimates how hard it is to know what poll responders actually think in a context of social pressure, see the phenomenon of the "shy Trump voter" and look how hard a time Western poll specialists have had tracking actual support with full access to people and transparent processes.