\documentclass[convert]{standalone} ignores linebreak

By default the standalone class doesn't make paragraphs. However, with the varwidth option it does; so

\documentclass[varwidth,convert]{standalone}

will do what you want, adjusting the width to the longest line.

There are several possibilities:

Table

\documentclass{standalone}

\begin{document}
\begin{tabular}{@{}l@{}}
  line 1\\
  line 2
\end{tabular}
\end{document}

Remarks:

• A tabular adds \struts in the rows, that might cause small white margins.
• Class standalone only uses the character bounding boxes and does not know about the bounding boxes of the glyph appearance. Thus additional margin makes sense to avoid cut off characters.
• The result can be run through pdfcrop to remove the remaining white margins.

Package varwidth

The environment varwidth of the package with the same name is a kind of minipage that is horizontally shrunk afterwards:

\documentclass{standalone}
\usepackage{varwidth}

\begin{document}
\begin{varwidth}{\linewidth}
  line 1

  line 2
\end{varwidth}
\end{document}

Class standalone provides the option varwidth for this use case (thanks mozartstrasse for the hint). That simplifies the example:

\documentclass[varwidth]{standalone}

\begin{document}
  line 1

  line 2
\end{document}

Character clipping

The problem that the character glyphs are outside their official font bounding boxes cannot be solved inside TeX, because TeX does only knows the official character bounding boxes and not their visual appearances.

Example:

\documentclass[varwidth]{standalone}

\begin{document}
  \raggedleft
  \itshape
  line f

  line $\not$
\end{document}

Part of f is not visible and \not has vanished entirely (extreme example, because \not has width zero, it overlaps the following relational operator).

As workaround a larger margin can be added:

\documentclass[margin=10pt,varwidth]{standalone}

And the result is cropped, e.g. via pdfcrop:

\documentclass[preview,multi]{standalone}

\begin{document}
\preview
line 1
\endpreview

\preview
line 2
\endpreview
\end{document}