Under the assumption your "big file" doesn't change that often: create an index of seek/tell values per line.

Sample code without error checking:

open IN, "bigfile.txt";
open  IDX, ">bigfile.idx";
binmode IDX;
print IDX, pack "N", 0;
while(<IN>) {
    print IDX pack "N", tell IN;
}
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Now, you can quickly locate line 10000 (the very first line has index 0):

my $lineno = 10000;

open IN, "bigfile.txt";
open IDX, "bigfile.idx";
binmode IDX;
seek IDX, 4 * $lineno, 0;
sysread IDX, (my $buffer), 4;
seek IN, unpack("N", $buffer), 0;
my $line = <IN>;
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This will work as long as your file length of the big file fits into 32 bits.