Q:

Are closed convex subsets of a Banach space weakly closed without the axiom of choice? 

It is a well-known fact that closed convex sets in Banach spaces are weakly closed.  The common proof is based on the Hahn-Banach theorem that uses the axiom of choice.  Is there any proof of this fact that avoids the axiom of choice and uses only ZF? 

A:

Of course not. 
The Hahn–Banach is equivalent to the assertion that $X^*$ is nontrivial for any nontrivial Banach space (or normed space, if you prefer). 
This means that if HB fails, there is a nontrivial Banach space $X$ whose weak topology is indiscrete, and in particular no set (other than $X$ and $\varnothing$) is weakly closed.