As stated in previous answers, the answer to the question is no. Here is another viewpoint, which gives additional information about the relationship between $G$ and $BG$:

A pair $(G,BG)$ where $G$ is a topological group homotopy equivalent to a finite CW complex, and $BG$ its classifying space, is called a *finite loop space* in the literature. So the question is if a finite loop space $G$ can have classifying space $BG$ homotopy equivalent to a finite CW complex, and the answer to this is no.

As pointed out in the previous answers, to get that $BG$ does not have a finite dimensional model, it is easy to reduce to the case where $G$ is *connected* (using that $BG$ is not finite for $G$ a finite group), so let's assume this.

Now, by the structure of Hopf algebras (Milnor-Moore) $H^*(G;{\mathbb Q})$ is an exterior algebra on a number $r$ of odd dimensional generators. The number $r$ is called the (rational) *rank*, and agrees with the usual notion of rank of compact Lie groups. Hence by a spectral sequence argument

$H^*(BG;{\mathbb Q})$ is a polynomial algebra on $r$ generators.

So the non-finiteness of $BG$ is a corollary of the following well-known fact:

**Fact:** A non-contractible connected finite loop space (or even H-space) has positive rank $r$.

*Proof of Fact:* The claim e.g., follows from a more general statement saying the rank is also equal to the number of odd degree generators for the mod $p$ cohomology (see Kane: Homology of Hopf Spaces Section 13-3, which uses the Bockstein spectral sequence to deduce this). To get the more limited statement of the rank being positive one can give a more pedestrian argument: Suppose that $G$ is non-contractible. Then (since its a simple space and a CW complex) $\tilde H^*(G;{\mathbb Z}) \neq 0$. If this cohomology is torsion free, then the rational cohomology is non-trivial, and we are done. So suppose there is torsion. If there is non-trivial $p$-torsion in $\tilde H^*(G;{\mathbb Z})$ then, by the universal coefficient theorem, $H^*(G;{\mathbb F_p})$ has cohomology in two consecutive degrees. So $H^*(G;{\mathbb F_p})$ has to have generators as a ring in odd degrees. But then the structure of Hopf algebras algebras shows that the Euler characteristic of $G$ has to be zero (since $H^*(G;{\mathbb F_p})$ contains an exterior tensor summand on a odd degree class, if $p$ is odd, and a truncated polynomial algebra on an odd degree class, truncated at $x^{2^k}$ for some $k$, if $p=2$). But $\chi(G) =0$ implies $\tilde H^*(G;{\mathbb Q}) \neq 0$ as wanted.

**Additional info:** There has been a lot of work on finite loop spaces, starting with the work of Hopf and Serre in the 1940's and 1950's, since they are generalizations of compact Lie groups. (Hopf looked at the weaker notion of H-spaces.) There is by now in fact a classification of (connected) finite loop spaces; see Section 3 of my survey paper http://www.math.ku.dk/~jg/papers/icm.pdf