Command that returns current font size as a length?

\makeatletter
\newcommand*\fsize{\dimexpr\f@size pt\relax}
\makeatother

This way you can use the macro like \setlength{\baselineskip}{1.33\fsize}.

\selectfont is the switch the changes the font. You can tie into this and update \fsize:

\documentclass{article}

\usepackage{letltxmacro}

\newlength{\fsize}
% https://tex.stackexchange.com/q/88001/5764
\LetLtxMacro\oldselectfont\selectfont
\makeatletter
\DeclareRobustCommand{\selectfont}{\oldselectfont\setlength{\fsize}{\f@size pt}}
\makeatother

\begin{document}

\tiny\the\fsize

\scriptsize\the\fsize

\footnotesize\the\fsize

\small\the\fsize

\normalsize\the\fsize

\large\the\fsize

\Large\the\fsize

\LARGE\the\fsize

\Huge\the\fsize

\end{document}

If you set the length, it won't change until you reset it. What you need is a macro:

\makeatletter
\newcommand{\currentfsize}{\f@size pt}
\makeatother

Example (using the code by Werner):

\documentclass{article}

\makeatletter
\newcommand{\currentfsize}{\f@size pt}
\makeatother

\newdimen\fsize
\newcommand{\setfsize}{\setlength{\fsize}{\currentfsize}}

\begin{document}

\setfsize % should be 10pt

\tiny\currentfsize

\scriptsize\currentfsize

\footnotesize\currentfsize

\small\currentfsize

\normalsize\currentfsize

\large\currentfsize

\Large\currentfsize

\LARGE\currentfsize

\Huge\currentfsize

\the\fsize % should be 10pt (printed in \Huge size)

\end{document}

As you see, setting \fsize at the beginning doesn't change its value.