This will do the job, provided you only want these symbols in text and display style, not in scriptstyle og scriptscriptstyle. If you do, then we'd have to work a bit harder:
\documentclass[]{article}
\newcommand\smallmath[2]{#1{\raisebox{\dimexpr \fontdimen 22 \textfont 2
- \fontdimen 22 \scriptscriptfont 2 \relax}{$\scriptscriptstyle #2$}}}
\newcommand\smalloplus{\smallmath\mathbin\oplus}
\newcommand\smallotimes{\smallmath\mathbin\otimes}
\begin{document}
\noindent
$ a \oplus b \otimes c$\\
$ a \smalloplus b \smallotimes c$
\end{document}

Explanation: The \fontdimen 22 parameters denote the axis height of the math font. As we set the symbol in scriptscriptstyle inside a textstyle setting, we need to raise the symbol by the difference of the two axis heights. Note that \dimexpr requires eTeX; if your LaTeX is not using eTeX, your TeX installation is very old and should be upgraded.
Of course, you may opt for replacing all the \scriptscript… by \script…, if this is too extreme.
scalebox? See https://tex.stackexchange.com/questions/13460/scalebox-knowing-how-much-it-scales (second answer) – Raven Sep 15 '18 at 13:55