How to enable use of minus and plus in tex documents?

To perform simple arithmetic operations with lengths you can use the e-TeX primitive \dimexpr:

\documentclass{article}

\newlength{\lengtha}
\setlength{\lengtha}{.5\textwidth}
\newlength{\lengthb}
\setlength{\lengthb}{.5\textwidth}

\begin{document}

\noindent\begin{tabular*}{\dimexpr\lengtha+\lengthb\relax}{cc}
\hline
a & b \\
\hline
\end{tabular*}

\end{document}

For further details, please refer to the e-TeX manual.

Something different is the use of plus and minus to specify glue in TeX:

<dimen>plus<dimen>minus<dimen>

where the plus<dimen> and minus<dimen> are optional and set to zero if not present; plus introduces stretchability, minus introduces shrinkability.

Use the calc package. Package description:

Adds infix expressions to perform arithmetic on the arguments of the LaTeX commands \setcounter, \addtocounter, \setlength, and \addtolength.

\documentclass{article}

\usepackage{calc}

\newlength{\lengtha}
\setlength{\lengtha}{.5\textwidth}
\newlength{\lengthb}
\setlength{\lengthb}{.5\textwidth}

\begin{document}

\begin{tabular*}{\lengtha + \lengthb}{cc}
\hline
a & b \\
\hline
\end{tabular*}

\end{document}