In platzkart there are two places where you can put your luggage — below the lower bunk and on the "third" (luggage) bunk above the second bunk. There is no rule that luggage of both passengers should fit below the lower bunk; on contrary, there is a rule that the passenger travelling on the lower bunk has the priority to use the space below their bunk, and the passenger on the top bunk has the priority to use the luggage bunk (see here, in Russian).
The luggage bunks are really spacious and you can fit rather big items there. It is quite common for tourists to put there their bicycles or canoes (disassembled, but still very bulky). It is even common, when a group of tourists occupy the whole compartment, to put bulky items across the carriage so that they span from one luggage bunk to another. (See this photo of several baidarkas (a two- or three-person canoe) lying on the third bunk.) I would estimate (from memory) the size of a single bunk as 180 x 60 centimeters, and the combined size of two bunks with space between them as 180x180 cm.
The official regulations (see link above) say that each passanger's luggage should not exceed 36 kg in weights and 180 cm in summary dimensions, but this rule is never enforced as long as there is no conflicts between passengers themself, and tourists commonly carry much bigger items.
As for separate luggage cars, from what I know, the common passenger trains do not contain them, but there are separate luggage trains. You can send your luggage on them, but they are very slow and you will have to pack your luggage very strongly, and not expect delicate handling.
However, there are special rules for luggage transportation that say that for each passenger ticket you can carry up to three items of luggage not exceeding 180 cm in summary dimensions, and not exceeding 75 kg each and 200 kg total. It not quite clear though where they would transport it, and I have no experience with it.
Also, there is a rule (see the first link above) that you can carry up to 50 kg more of luggage in standard carriage if you buy an additional ticket (a standard passenger ticket, apparently, though it's not quite clear).