Let $k$ be a number field, and $F/k$ a finite extension. I would like to find a countable family of extensions $k_i/k$ of degree 2 and a place $v_i$ of $k_i$ such that if $v$ is the place of $k$ lying below $v_i$, then $[k_{v_i}:k_v]$=2, where $k_{v_i}$ and $k_v$ are the completions of $k_i$ and $k$ at $v_i$ and $v$. Furthermore, for some place $w$ of $F$ lying above $v$, I want the $F_w = k_v$. Is this possible?

There are infinitely many places $v$ of $k$ such that $F_w=k_v$ for some $w$ above $k$. For each such place, take $a_v \in k, v(a_v)=1$ and consider the extension $k(\sqrt{a_v})/k$. It has the property you want at $v$ and you will get infinitely many such extensions as you vary $v$, so you are done.