Nope. It means that you must interpret the sheaf $\cal{F}$ as a complex concentrated in zero degree (if $\cal{F}$ is a single sheaf). Maybe a "picture" could help:
$$
\begin{array}
{}\vdots & & \vdots \\
\uparrow & & \uparrow \\
0 & \rightarrow & C^2\cal{F} \\
\uparrow & & \uparrow \\
0 & \rightarrow & C^1\cal{F} \\
\uparrow & & \uparrow \\
\cal{F} & \rightarrow & C^0\cal{F}
\end{array}
$$
But, if $\cal{F}^\bullet$ is also a complex of sheaves, then $C^\bullet (\cal{F}^\bullet)$ is a double complex. And the quasi-isomorphism is between the complex $\cal{F}^\bullet$ and the total complex $\mathrm{Tot}\ C^\bullet (\cal{F}^\bullet)$ of that double one; that is the (simple) complex which its $n$ degree is
$$
\bigoplus_{p+q=n} C^p\cal{F}^q \ .
$$
The corresponding "picture":
$$
\begin{array}
{}\vdots & & \vdots & & \vdots \\
\uparrow & & \uparrow & & \uparrow \\
\cal{F}^2 & \rightarrow & C^0\cal{F}^2 & \rightarrow & C^1\cal{F}^2 & \rightarrow & \dots \\
\uparrow & & \uparrow & & \uparrow \\
\cal{F}^1 & \rightarrow & C^0\cal{F}^1 & \rightarrow & C^1\cal{F}^1 &\rightarrow & \dots \\
\uparrow & & \uparrow & & \uparrow \\
\cal{F}^0 & \rightarrow & C^0\cal{F}^0 & \rightarrow & C^1\cal{F}^0 & \rightarrow & \dots
\end{array}
$$
This is true at least when the complex of sheaves $\cal{F}^\bullet$ is concentrated in positive degrees. Otherwise, you would need some finite cohomological dimension hypothesis on your topological space (or Grothendieck site) where your sheaf is defined.