I would not say the Russian state itself, but some Cossacks leased some land in Djibouti from an Afar sultan and then came by ship to set up a colony. They named it New Moscow, and it only managed to exist for a single month before getting driven out by the French.

The Russian American Company under Baranov (which was essentially a state-owned monopoly commercial company) not only established bases in Alaska, but founded Fort Ross Colony in California, and at various times was negotiating with King Kahmehameha of Hawaii for development of Russian-owned farms there. Lots of references, including Black, Lydia T. "Russians in Alaska 1732-1867", U of Alaska Press 2004, AND Bancroft, Hubert H. "History of Alaska 1730-1885" Reprint Antiquarian Press 1959. Note that in some areas of his work, Bancroft is derided, but his references on the Hawaiian connection seem to be accurate.