Proteins are constantly being synthesized in a living cell. Why doesn't the number of protein molecules become too great for the cell to contain, leading to cell destruction?

Both eukaryotes and prokaryotes have gene regulation strategies. By gene regulation, the amount of protein is maintained in the living cell.

In prokaryotes, operons are present which are controlled by a regulatory gene that regulates the synthesis of protein in prokaryotes. In eukaryotes, repressor and transcription activators regulate the expression of gene and protein formation.

So by gene regulation cell maintains the number of protein molecules to become too great for a cell. Without gene regulation the survival of any organism is not possible.