Ok, so I try to formulate rigorously the question in the title, for which I am asking for references. My definitions may be flawed, so feel free to adjust/correct them! I care about dimensions 2 and 3 below, but feel free to mention general d as well.

Let $d\ge 1$ be an integer, $G=(V,E)$ be a graph, $w:E\to [0, \infty)$ be a weight function. A realization in $\mathbb R^d$ with graph $G$ and weight $w$ is a map $P:V\to\mathbb R^d$ with the further property that $|P(v)-P(v')|=w(v,v')$ whenever $\{v,v'\}\in E$. I will identify such $P$ with its image, I hope it's not a problem.

**Edit** (May 11, 2020)**:**
*As pointed out by Misha, this below definition is not correct. The action of isometries makes the set of all realizations of a linkage always cover all $\mathbb R^d$. He indicates a paper in which a more inclusive definition is formulated in $d=2$.*

(Previous "wrong" definition:

I say that a set $A\subset \mathbb R^d$ is realizable by linkages if there exists $G,w$ as above and an cover of $A$ by open sets of $\mathbb R^d$ such that for every $U\subset\mathbb R^d$ in the cover there exist $G,w$ such that that the union of all (images of) realizations of $G,w$ in $\mathbb R^d$ intersected with $U$ coincides with $A\cap U$.)

*To "fix the problem", following the we will allow a subset of vertices of $G$ to be kept fixed in $\mathbb R^d$. In dimension $2$ this apparently generalizes the definition in the above paper, but I think that the result of the paper still allows to reply positively to the $d=2$ case of the question, with little extra work.*

**Revised definition:** We say that $A\subset \mathbb R^d$ is realizable by linkages if there exists a cover of $A$ by open sets of $\mathbb R^d$ such that for every $U\subset\mathbb R^d$ in the cover there exist $G=(V,E)$ and $w$ as above, a subset $F\subset V$, and a map $\phi: F\to\mathbb R^d$, such that the union of all (images of) those realizations of $G,w$ which restricted to $F$ equal $\phi$, intersected with $U$, coincides with $A\cap U$.

**Question**: Say $d=2$ or $d=3$. Is it true that all algebraic sets $A\subset\mathbb R^d$ are realizable by linkages? What are references for this?

*(Note: as of May 11 2020, it appears to me that the case $d=2$ is nicely treated in the answers given, while the case $d=3$ is not yet treated, possibly due to the previously bad definition.)*

I found some mention of this, without references on Branko Grünbaum's "Lectures on lost mathematics", dated around 1975, and he says there that $d=2$ case is known, but does not give references, and $d=3$ case is a question by Hilbert which is open (but again no references there).