A Russian saying, otdat’ poslyednyuyu rubashku (aht-daht’ pahs-lyed-nyu-yu roo-bahsh-koo; to give away one’s last shirt), makes the point that you have to be giving, no matter what the expense for yourself. In Russia, offering guests whatever they want is considered polite. Those wants don’t just include food or accommodations; old-school Russians offer you whatever possessions you comment on, like a picture on the wall, a vase, or a sweater.
Now, being offered something doesn’t necessarily mean you should take it. Russians aren’t offering something because they want to get rid of it; they’re offering because they want to do something nice for you. So, unless you feel that plundering their home is a good idea, don’t just take things offered to you and leave. Refuse first, and do so a couple of times, because your hosts will insist. And only accept the gift if you really want this special something, but then return the favor and give your hosts something nice, as well.