table rescaling and subcaption package

You could (i) use the tabular* instead of the tabular (or tabularx) environment, (ii) set the width of the tabular* to \linewidth, and (iii) use the (admittedly complicated-looking) expression @{\extracolsep{\fill}} to make LaTeX expand the intercolumn white space so that the contents of the table take up the full available width. (Unless instructed to do something else, LaTeX inserts intercolumn whitespace in the amount of 2\tabcolsep between columns in a tabular... environment.)

Separately, I'd also get rid of the \large font size directive, use @{} to get rid of the extra whitespace at the left and right margins of the tables, not use \centering and instead use \hspace*{\fill} to separate the two subtables, and make a few other changes here and there to streamline the setup and look of the tables.

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Addendum, posted after receiving information that the OP uses a fairly narrow text block: It's still possible to use the tabular* environment, but with two modifications relative to the code I posted originally (now deleted): (i) reduce the value of the \tabcolsep parameter from the default value of 6pt to 1pt and let LaTeX then insert \fill for extra separation, and (ii) change the font size to \small (which is 1pt less than the default font size).

\documentclass{article}
\usepackage{siunitx,booktabs,subcaption}
\setlength\tabcolsep{1pt}  % LaTeX default is 6pt
\sisetup{detect-weight=true,
    detect-inline-weight=math,
    table-format=1.2(2)}
\begin{document}
\begin{table}
\small % reduce font size by 1pt
     \begin{subtable}{0.48\textwidth}
     \begin{tabular*}{\linewidth}{@{}l 
        @{\extracolsep{\fill}} SS
        S[table-format=2.2(2)]
        S[table-format=2.2(2)]@{}}
          \toprule
          \phantom{Var.} &  
          \multicolumn{4}{c}{Error Quantiles}\\
          \cmidrule{2-5}
          &  {25\%} & {50\%} & {75\%} & {95\%}\\
          \midrule
          A  & 0.17(1) & 0.25(3) & 0.33(2) & 0.43(2)\\
          B  & 0.09(1) & 0.12(1) & 0.17(2) & 0.26(5)\\
          C  & 0.07(1) & 0.09(1) & 0.11(1) & 0.16(1)\\[1ex]
          A & 0.17(83) & 0.26(83) & 0.38(83) &  0.96 \\ % 1e7
          B & 0.10(1)  & 0.16(1)  & 0.22(2)  &  0.41(8)\\
          C & 0.08(1)  & 0.12(1)  & 0.19(3)  &  0.52(23)\\[1ex]
          A & 0.37(6) & 0.73(15) & 1.66(63) & {201}\\ % 1e5
          B & 0.07(1) & 0.09(1)  & 0.15(2)  &  15.50(33)\\
          C & 0.06(0) & 0.09(1)  & 0.12(1)  &   0.20(2)\\
          \bottomrule
     \end{tabular*}%
     \caption{}
     \end{subtable}%
     \hspace*{\fill}%
     \begin{subtable}{0.48\textwidth}
     \begin{tabular*}{\linewidth}{@{}l 
        @{\extracolsep{\fill}} SS
        S[table-format=2.2(2)]
        S[table-format=2.2(2)]@{}}
          \toprule
          \phantom{Var.}
          &  \multicolumn{4}{c}{Error Quantiles}\\
          \cmidrule{2-5}
          &  {25\%} & {50\%} & {75\%} & {95\%}\\
          \midrule
          A  & 0.17(1) & 0.25(3) & 0.33(2) & 0.43(2)\\
          B  & 0.09(1) & 0.12(1) & 0.17(2) & 0.26(5)\\
          C  & 0.07(1) & 0.09(1) & 0.11(1) & 0.16(1)\\[1ex]
          A & 0.17(83) & 0.26(83) & 0.38(83) &  0.96 \\ % 1e7
          B & 0.10(1)  & 0.16(1)  & 0.22(2)  &  0.41(8)\\
          C & 0.08(1)  & 0.12(1)  & 0.19(3)  &  0.52(23)\\[1ex]
          A & 0.37(6) & 0.73(15) & 1.66(63) & {201}\\ % 1e5
          B & 0.07(1) & 0.09(1)  & 0.15(2)  &  15.50(33)\\
          C & 0.06(0) & 0.09(1)  & 0.12(1)  &   0.20(2)\\
          \bottomrule
     \end{tabular*}%
     \caption{}
     \end{subtable}
\end{table}
\end{document}

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Second Addendum: In case you feel that this setup (with font size \small, 10% smaller than the "normal" font size) doesn't provide enough whitespace between the columns, you could use \footnotesize (20% smaller than the "normal" font size):