Your professor should know better. This teaches nothing about LaTeX, in my opinion; using packages is the way.
But here it is; it includes a (perhaps too clumsy) implementation of Euclid's algorithm for the greatest common divisor, in order to pass a correct pair of values to \line.
\documentclass{article}
\makeatletter
\newcommand{\crossout}[1]{%
\begingroup
\settowidth{\dimen@}{#1}%
\setlength{\unitlength}{0.05\dimen@}%
\settoheight{\dimen@}{#1}%
\count@=\dimen@
\divide\count@ by \unitlength
\count0=20 \count4=\count@
\loop
\count2=\count0 % keep a copy
\divide\count2\count4 \multiply\count2\count4
\ifnum\count2<\count0
\advance\count0 -\count2 % the remainder
\count2=\count0
\count0=\count4
\count4=\count2
\repeat
\count0=20 \divide\count0\count4
\count2=\count@ \divide\count2\count4
\begin{picture}(0,0)
\put(0,0){\line(\count0,\count2){20}}
\put(0,\count@){\line(\count0,-\count2){20}}
\end{picture}%
#1%
\endgroup
}
\makeatother
\begin{document}
This word is \crossout{crossed} out
\end{document}

Of course, presenting this solution would be cheating. And, no, it won't work in all cases, due to strict limitations on the pairs accepted by \line. Your professor surely knows that TeX doesn't draw oblique lines.
With the standard package pict2e it's easier and it will work in any case.
\documentclass{article}
\usepackage{pict2e}
\makeatletter
\newcommand{\crossout}[1]{%
\begingroup
\settowidth{\dimen@}{#1}%
\setlength{\unitlength}{0.05\dimen@}%
\settoheight{\dimen@}{#1}%
\count@=\dimen@
\divide\count@ by \unitlength
\begin{picture}(0,0)
\put(0,0){\line(20,\count@){20}}
\put(0,\count@){\line(20,-\count@){20}}
\end{picture}%
#1%
\endgroup
}
\makeatother
\begin{document}
This word is \crossout{crossed} out
\crossout{word}
\crossout{U}
\end{document}

A different solution (pdftex or luatex only)
\documentclass{article}
\makeatletter
\newcommand{\crossout}[1]{%
\begingroup
\sbox\z@{#1}%
\dimen\z@=\wd\z@
\dimen\tw@=\ht\z@
\dimen\z@=.99626\dimen\z@ % get big points
\dimen\tw@=.99626\dimen\tw@ % get big points
\edef\co@wd{\strip@pt\dimen\z@}% just the number
\edef\co@ht{\strip@pt\dimen\tw@}% just the number
\leavevmode
\rlap{\pdfliteral{q 1 J 0.4 w 0 0 m \co@wd\space \co@ht\space l S Q}}%
\rlap{\pdfliteral{q 1 J 0.4 w 0 \co@ht\space m \co@wd\space 0 l S Q}}%
#1%
\endgroup
}
\makeatother
\begin{document}
This is \crossout{crossed} out
\end{document}

tikzis a package. I've also deleted the 'color' tag. – Mico Apr 26 '15 at 22:57cancel.styyourself?!?! – kahen Apr 26 '15 at 22:59\documentclass{...}\usepackage{....}\begin{document}...\end{document}. If possible, it should compile and have the minimum amount of code needed to illustrate your problem. This makes it much easier for people to help you - and much more likely that they will! – Apr 26 '15 at 23:06\usepackage{cancel} \begin{document} %BeginDocument
%End Document \end{document}
– Sarah Apr 26 '15 at 23:09