MongoDB is fine at the conceptual level. The problem is that the architecture and implementation are consistently poor in myriad ways. It is a case study of what can happen when well-meaning individuals with little experience in database architecture and implementation attempt to build a scalable database engine.
That said, a competently engineered RDBMS can do everything NoSQL databases can, particularly limited databases like MongoDB. The caveat is that you have to learn how to use those databases; they are very feature rich and powerful but that flexibility makes them more complicated. PostgreSQL is a very good choice from the open source world and is just as fast as NoSQL du jour in the hands of someone that knows it.
I currently design extreme-scale real-time analytical database engines, so I have no vested interest in any particular solution (we are not really competing with the current market). If I was going to build a large-scale web app today and needed a backing database, I would go with PostgreSQL -- it is very capable and well-engineered.
...a competently engineered Turing Machine can do everything NoSQL databases can, particularly limited databases like MongoDB. ... Emacs is just as fast as NoSQL du jour in the hands of someone that knows it.
FTFY.
The difference between using a good competently engineered distributed database and PostgreSQL is that when your prime concerns are horizontal scaling and operations costs, the distributed database can be an order of magnitude simpler and faster than PostgreSQL given the same amount of effort.
