Can cell exist without Ribosomes?
The most essential thing for a cell to survive is a membrane. Without a membrane there is no boundary between cell and its surroundings. A selectively permeable membrane is needed for cells (from single celled organisms to elephants and giraffes and whales) to survive. Cells will die quickly on removal of cell membrane.

The RBCs are good example of cells without mitochondria or ribosomes which survive for 120 days.



Please read the article on Erythropoiesis for detailed information on how RBCs form and how they have proteins in their surface.

In short, the RBCs acquire the surface proteins during their formation period in the bone marrow. Inside the bone marrow the premature RBCs have nucleus, ribosomes, mitochondria, etc... and produce all the required structural and functional proteins. On maturing they loose the nucleus and cell organelles and contain only hemoglobin in their cytoplasm.