Drawing lines within circle using TikZ

Using clipping? I am using a scope here to restrict the clipping to it.

\documentclass[margin=5pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
    \begin{scope} 
        \draw  [clip] (0,0) circle (1cm);
        \draw (0,-1) -- (0,1); % first (center) line 
            \foreach \x in {1,...,4} {% 4 lines per side 
                \draw (\x/10, -1) -- (\x/10,1); % positive x position 0.1, 0.2...
                \draw (-\x/10, -1) -- (-\x/10,1); % and negative ones 
        }
    \end{scope}
\end{tikzpicture}
\end{document}

(Notice that the seemingly bigger line at the center is an aliasing problem of the PDF viewer).

Using scopes let you manipulate the thing quite easily:

\documentclass[margin=5pt]{standalone}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}
    \begin{scope}
        \draw  [clip] (0,0) circle (1cm);
        \draw (0,-1) -- (0,1); % first diagonal
            \foreach \x in {1,...,4} {% 4 lines 
                \draw (\x/10, -1) -- (\x/10,1);
                \draw (-\x/10, -1) -- (-\x/10,1);
        }
    \end{scope}
    \begin{scope}[xshift=2cm, scale=0.5, rotate=45]
        \draw  [clip] (0,0) circle (1cm);
        \draw (0,-1) -- (0,1); % first diagonal
            \foreach \x in {1,...,4} {% 4 lines 
                \draw (\x/10, -1) -- (\x/10,1);
                \draw (-\x/10, -1) -- (-\x/10,1);
        }
    \end{scope}
\end{tikzpicture}
\end{document}

For fun, here is a style vlines using path picture and grid to draw the lines.

\documentclass[tikz,border=7pt]{standalone}
\tikzset{
  vlines/.style={
    path picture={
      \draw[xstep=#1, ystep=100cm, shift={(path picture bounding box.south west)} ]
      (path picture bounding box.south west) grid (path picture bounding box.north east);
    }
  }
}
\begin{document}
  \begin{tikzpicture}
    \draw[vlines=2mm] (0,0) circle (1);
  \end{tikzpicture}
\end{document}

Note : The ystep=100cm can be set to the height of path picture bounding box but this is another story ;)