None of these animals can live in the same cage. As others have said rats are omnivores and will kill animals smaller them themselves.
Gerbils are extremely territorial and will often fight other gerbils to the death over territory(much less other rodents). I have a gerbil whom I rescued, who had it entire back ripped apart because of territorial cage mates. Gerbils can even decide to de-clan with cage mates they have been living with for months or years, causing fighting and territory issues
Syrian hamsters are also territorial, and prefer to live alone, and will kill cage mates(even brothers/sisters eventually). This type of hamster should never be housed with anything, even another hamster.
If you do not have the room for another pet, than it is better to stick with what you already have. You do not NEED another animal you WANT it. If you have done adequate research that you should come to the conclusion that not having enough room for your pets means you are not ready to own anything else.