This uses a variation on my censor package to accomplish the strikeout, since the \xblackout macro of that package can work across paragraph boundaries (but not across math, which the OP clarified is not needed).
To make the \xblackout macro operate as an environment, I used the environ packages feature of \BODY. Even so, I had to go through an intermediate macro \soutrefunexp in order to expand \BODY once, to make it digestible to \xblackout.
\documentclass{article}
\usepackage{environ,censor,calc}
\NewEnviron{mysout}
{\soutrefunexp{\BODY}}
%%%%%%%%%%%
\censorruleheight=.1ex %THICKNESS OF CENSOR RULE
\newlength\nextcharwidth
\makeatletter
\renewcommand\@cenword[1]{%
\setlength{\nextcharwidth}{\widthof{#1}}%
\censorrule{\nextcharwidth}%
\kern -\nextcharwidth%
#1}
\makeatother
\newcommand\soutref[1]{\censorruledepth=.55ex\xblackout{#1}}
\newcommand\soutrefunexp[1]{\expandafter\soutref\expandafter{#1}}
%%%%%%%%%%%
\begin{document}
What is this?
Another paragraph.
\begin{mysout}
What is this?
Another paragraph.
\end{mysout}
What is this?
Another paragraph.
\end{document}

Replacing \xblackout with \blackout produces the following slightly different result:
