It took me ages to find this reference (and I actually found it in this post). There is a theorem by Acquistapace–Broglia–Tognoli in *An embedding theorem for real analytic spaces* stating:

**Theorem.** Let $X$ be a paracompact connected $n$-dimensional analytic space and suppose that $q := \sup_{x \in X} \dim T_x X < \infty$. Then $X$ admits a closed $C^\omega$-immersion into $\mathbb{R}^{n+q}$.

This is a generalization of Grauert's embedding theorem for real analytic manifolds. From this we obtain:

**Corollary.** Let $X$ be a separated complex algebraic variety. Then $X^{\mathrm{an}}$ admits a topological closed embedding into $\mathbb{R}^N$ for some $N$.

Proof. $X^{\mathrm{an}}$ is a paracompact complex (thus also real) analytic space with finitely many connected components and with bounded tangent space dimensions. Now, apply the theorem.

Remark 1. The only thing that surprises me is why the ABT paper has only one citation on MathSciNet since 1979; I don't know if there's another source for the above theorem (I hope it's right actually).

Remark 2. I don't know if there isn't a simpler argument.