Put the word in a box, and measure the width of the box. For example,
\newdimen\stringwidth
\setbox0=\hbox{hi}
\stringwidth=\wd0
Note that if the quantity in the \hbox is a phrase, the actual
measurement only approximates the width that the phrase will occupy in
running text, since the inter-word glue can be adjusted in paragraph
mode.
The same sort of thing is expressed in LaTeX by:
\newlength{\gnat}
\settowidth{\gnat}{\textbf{small}}
This sets the value of the length command \gnat to the width of "small"
in bold-face text.
This question on the Web: http://www.tex.ac.uk/cgi-bin/texfaq2html?label=findwidth