After trying some other approaches, I finally create a Perl program to extract lines from the middle of a file. Just give it the line number to start at, and a second line number to stop at, and this Perl program will print all the lines from the file in that range.
The source code for this Perl program is shown below.
Sample usage:
extract.pl 500 1000 myBigFile > smallerFile
Here's the code for extract.pl: