The Metaculus crowd thinks that it's very likely that Russia will restrict emigration this month:

https://www.metaculus.com/questions/10080/will-russia-close-borders-by-2022-04-01/

(Just hover over the link to see the current prediction.)

I myself am more optimistic about this question resolving negatively. But even if Russia would not close borders for a notable fraction of its citizens, it seems fairly likely that the aviation industry will collapse or shrink due to sanctions.

Related forecasts:

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Related news from NYT

Russia’s state airline, Aeroflot, said on Saturday that it would suspend all international flights from Tuesday “due to additional circumstances that prevent the performance of flights.”

[...] Separately, Russia’s aviation authority, Rosaviatsiya, recommended on Saturday that Russian airlines with planes registered in foreign jurisdictions suspend all flights abroad from Sunday because of fears that they could be seized by foreign governments.

The suspension effectively leaves a handful of Western companies with no way of recovering hundreds of planes leased to Russian carriers. [...] According to IBA, an aviation consulting firm, there are nearly 600 foreign-owned planes currently leased to airlines in Russia. Those planes are worth an estimated $12 billion, about a third of which belongs to Irish lessors.

Border could be closed as soon as Friday when the Federation Council hosts a special meeting. 
 

Here is a document (in Russian) with lots of useful information about emigrating from Russia right now.  I agree people currently in Russia should strongly consider leaving. I am reasonably optimistic that European/etc governments will open pathways for Russians to apply for political asylum or similar in the nearish future. At the moment, Republic of Georgia might be a good option as Russians have almost 1 year of visa-free residence there.

I'd like to, but your governments have already closed borders for Russians.

Yeaaaah. Exactly. And not that it was too easy to move to a less sinister place even earlier.