How do red blood cells survive without a nucleus?
As explained in the previous answer, the RBC loses its nucleus only at maturity. The nucleus contains the DNA and which can in turn produce protein. No nucleus means - no protein/ mRNA synthesis. Also, obviously, the cell loses its capability to divide. 

Hemoglobin too is a protein. Knowing that it would lose its nucleus, hemoglobin is synthesized in the precursor stages of RBC. A mature RBC can not synthesize new Hb. Hence, the RBC keeps ready its store of enzymes and structural proteins (not only Hb, all proteins it would need for normal functioning) since it can not form new stuff once it has lost its nuclues.

The lack of a nucleus also limits the cell's repair capabilities. So human RBCs are cleared from circulation once they are about 4 months old. Damage (mostly from oxidative stress) to their structural proteins leads to loss in membrane flexibility. Since, the RBC can't replace the stuff, it gets gulped in by macrophages.

It is not surprising that the cell doesn't have a nucleus. The loss of a nucleus renders it better oxygen carrying capability. Even platelets donot have a nucleus. There too, enzymes are prepared and stored beforehand. 

Hope that helps!