Can a guinea pig and chinchilla share the cage?

Absolutely not. If you are talking about Syrian hamsters and mongolian gerbils then they are both incredibly territorial and would likely kill each other. Syrian hamsters are solitary and cannot even be kept with other Syrians.

Mongolian gerbils do well in pairs or groups of their own kind, but you cannot even introduce a lone adult to a lone adult without a splitcage as they are highly territorial and would fight an intruder to the death.

Hamsters and gerbils also need different types of cages. It is vital that a gerbil has a tank as part of its cage since they need plenty of room and substrate to dig whereas a hamster cannot be kept in a tank due to the ventilation and health problems it would cause.

Cohabiting different types of rodent should only rarely be attempted and only be an experienced rodent keeper. There are only a very few examples where it can work, and the vast majority of cases you would just end up with dead pets.