Which are the best quartiers (areas) in Moscow to stay for a 4-day visit?
I would recommend to stay near the city center, close to the Krasnaya ploshad (Red Square). This way you have a lot of sightseeing spots in walking distance.
We spent 3 days in Moscow recently and stayed at the City Home Hostel which was very convenient, and prices are ok!
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Just a tip: The closer to Kremlin you are, the more expensive it will be. I recommend Kitai-Gorod