I'd like the whole Verbatim (fancyvrb) block to have a background
color. How do I do it?
The fancyvrb package does define a fillcolor -- but that only colors
the area around the margin -- not the background of the block itself.
I'd like the whole Verbatim (fancyvrb) block to have a background
color. How do I do it?
The fancyvrb package does define a fillcolor -- but that only colors
the area around the margin -- not the background of the block itself.
Use package listings instead. It supports page breaks and also fancyvrb:
\documentclass{article}
\usepackage{listings}
\usepackage{xcolor}
\begin{document}
\begin{lstlisting}[backgroundcolor=\color{yellow},frame=single,basicstyle=\ttfamily]
\addcontentsline{toc}{chapter}{The first section name}
foo
\end{lstlisting}
\end{document}

You can use the very powerful tcolorbox package, in combination with fancyvrb
\documentclass{article}
\usepackage{tcolorbox,fancyvrb,xcolor,tikz}
\tcbuselibrary{skins,breakable}
\newenvironment{BGVerbatim}
{\VerbatimEnvironment
\begin{tcolorbox}[
breakable,
colback=yellow,
spartan
]%
\begin{Verbatim}}
{\end{Verbatim}\end{tcolorbox}}
\begin{document}
\begin{BGVerbatim}[numbers=left,numbersep=6pt]
abc def {
xyz
\end{BGVerbatim}
\end{document}
The numbers=left option has been added just to show that you can pass fancyvrb options to the new environment.

You can do that with the fancybox package, which defines the same Verbatim environment.
You can define a new verbatim environment, let's say MyVerbatim with background and text colors.
\newenvironment{MyVerbatim}{\VerbatimEnvironment%
\noindent\begin{Sbox}
\begin{minipage}{\dimexpr\linewidth-2\fboxsep-2\fboxrule}
\begin{Verbatim}
}{%
\end{Verbatim}%
\end{minipage}%
\end{Sbox}%
\fcolorbox{black}{yellow}{\TheSbox}%
}
Change black and yellow to whatever you like.
MWE:
\documentclass{article}
\usepackage{fancybox}
\usepackage{xcolor}
\newenvironment{MyVerbatim}{\VerbatimEnvironment%
\noindent\begin{Sbox}
\begin{minipage}{\dimexpr\linewidth-2\fboxsep-2\fboxrule}
\begin{Verbatim}
}{%
\end{Verbatim}%
\end{minipage}%
\end{Sbox}%
\fcolorbox{black}{yellow}{\TheSbox}%
}
\begin{document}
\begin{MyVerbatim}
Hello
I have a background!
\end{MyVerbatim}
\end{document}
Output:

You can use tcolorbox with listings package seemlessly. There by one can combine the advantages and features of both. Here is an example.
\documentclass{article}
\usepackage{tcolorbox,listings}
\lstdefinestyle{mystyle}{
basicstyle=\ttfamily,
numbers=left,
numberstyle=\tiny,
numbersep=5pt
}
\tcbuselibrary{listings,skins,breakable}
\newtcblisting{BGVerbatim}{
arc=0mm,
top=0mm,
bottom=0mm,
left=3mm,
right=0mm,
width=\textwidth,
boxrule=0.5pt,
colback=yellow,
spartan,
listing only,
listing options={style=mystyle},
breakable
}
\begin{document}
\begin{BGVerbatim}
abc def {
xyz
\end{BGVerbatim}
\end{document}

You can do this with the package fancyvrb defined the environment Verbatim. In my code, I use the invironment figure and especially apply \textcolor without affecting to the pre-formatting of the text inside Verbatim.
\documentclass{article}
\usepackage{fancyvrb,xcolor}
\begin{document}
\newcommand\hlca[1]{\textcolor{red}{\textbf{#1}}}
\newcommand\hlcb[1]{\textcolor{blue}{\textbf{#1}}}
\begin{figure}[h!]
\centering
\begin{SaveVerbatim}[commandchars=\\\{\}]{efms}
EFM1: \hlca{R9} R10i T1 \hlcb{T2}
EFM2: \hlca{R9} R10i R11i R12 R13 R14 T1 \hlcb{T5}
EFM3: R6i \hlca{R9} R10i R11i R12 R13 R14 \hlcb{R15} T1 T6 T7
EFM4: \hlcb{R6i} \hlca{R7i} R8i \hlca{R9} R10i R11i R12 R13 R14 T1 T6 T7
EFM5: \hlca{R7i} R8i \hlcb{R15} T1 T6
EFM6: \hlcb{R6i} \hlca{R7i} R8i T1 \hlcb{T5} T6 T7
EFM7: \hlcb{R6i} \hlca{R7i} R8i R11i R12 R13 R14 T1 \hlcb{T2} T6 T7
\end{SaveVerbatim}
\fcolorbox{black}{TeaGreen}{\BUseVerbatim{efms}}
\caption{List of 7 EFMs concerning the production of external citrate}
\label{box:7-efms-external-citrate}
\end{figure}
\end{document}
Below is the result when running the code.
.