I think I understand now
you want the text between the first Start and first Exit
you can throw in a last to exit the loop when matching Exit

open (DATA,"<xxx.txt") || die "Can't open xxx.txt: $!\n";
while(<DATA>) {
    if (/Start context/ .. /\* Exit context \*/)  {
         print unless /(Start context|\* Exit context \*)/;
         last if /\* Exit context \*/;
    }
}
[download]