Some nations may choose to extend extraterritoriality to foreign embassies on their soil, and some countries may require that the host country offer them extraterritoriality in exchange for setting up an embassy, but there's no requirement for it outside of those private agreements. The Vienna Convention on Diplomatic Relations actually states that embassies operate under the laws of their host countries (the nation they are located in), but simply prohibits the host country from entering or searching the embassy without permission. It also extends the same protections to the homes of the diplomats. This may create a de facto form of extraterritoriality (as in, the Russians can't enforce Russian law on the soil of foreign embassies), but the embassy doesn't actually become "foreign soil".
The host country also does retain the right to close an embassy or consulate at any time. There is nothing stopping the United States from informing Russia that its embassy will no longer be granted the status of a diplomatic mission, and will not be covered under the Vienna Convention. The files and contents of the building would retain their protected status as long as they were controlled by a diplomat, but the building itself would no longer be immune to entry. The Russian's would then have to decide whether to continue operating in the building without diplomatic protections, or whether to close the office and pull out.
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