Gnuplot was designed to quick plot preprocessed data.
Under Linux, however, you can take advantage of many ready tools to manipulate your data.
If your dataset is not huge you can choose to use some of them inline from gnuplot.
sort can be one of this (as well as awk) here you read 4 times your file.
plot "<sort data.dat -k2 -n | sort -n -k1 -u" t "Min" \
,"<sort data.dat -k2 -n -r| sort -n -k1 -u" t "Max"
With awk you can do a finer work scanning your file only one time
set xlabel "My x axis [a.u.]"
set ylabel "My y axis [a.u.]"
set style fill transparent solid 0.4
plot "< awk '{ Q[$1]++; \
if (Q[$1]==1){ Min[$1]=$2;Max[$1]=$2; } \
else \
{ if (Min[$1]>$2) {Min[$1]=$2;} \
else {if (Max[$1]<$2) Max[$1]=$2;} } \
} \
END {for (i in Min) print i,Min[i],Max[i]}' data.dat " \
using ($1):($2):($3) w filledcurve title "Min and Max" lc rgb "#00A000"

Notes:
- I purged the 1st line of your file for simplicity
plot "< command " it executes a command and it uses it as input data
man sort and man awk to have the full help of the sort and awk programs
- check associative arrays in awk to deeper understand
awk.