Localification of a ringed space Call a ringed space local it if it lies in the image of the obvious faithful, non-full functor from locally ringed spaces to ringed spaces. 
Given a ringed space, is there a map $f$ from it to some local ringed space such that any other map from it to a local ringed space much factor uniquely through $f$?
 A: So you are asking if the forgetful functor from locally ringed spaces to ringed spaces has a left adjoint. See here
A: This is too long for a comment. So, I am writing here.
Let $(X,\mathcal{O}_X)$ be a ringed space. The idea is to localize the ring $\mathcal{O}_x$ for each open $x\in X$. Let $\mathcal{L}_x$ denote the localization of $\mathcal{O}_x$. Now one can ask if this collection $\{\mathcal{L}_x\}_{x\in X}$ form a sheaf on $X$??
Is this what you are thinking??
The idea is to localize the rings $\mathcal{O}_x$ but there is no obvious choice of prime ideals where we can localize. So, I think we need to fix a prime ideal $P_x$ in $\mathcal{O}_x$ for each $x\in X$. Then, we localize $\mathcal{O}_x$ at $P_x$ and call this $\mathcal{L}_x$. I am almost sure this collection $\{\mathcal{L}_x\}_{x\in X}$  gives a sheaf on $X$.
Universal property will then be depending on the choice of prime ideals $\{P_x:x\in X\}$.
The paper Localization of Ringed spaces by W. D. Gillam is relevant here. I will add more details soon.
A: Let $\text{RS}$ be the category of ringed spaces, $\text{LRS}$ the category of locally ringed spaces. As you allude to, there is a forgetful functor $\mathcal{F}$: $\text{LRS} \rightarrow \text{RS}$, that is not full. As GLe notes, this functor has no left adjoint, since it does not preserve limits. An example is provided in the link GLe gave. From that example, I think you should be able to concoct an example of a fiber product (pullback), that isn't preserved. 
