The minimum height for an adult hitter is 8', but it's important to remember at that height an adult won't be able to throw full overhand. It's also important to remember that a taller cage will last longer because of the increased distance between batter and net.Taller cages also allow for better ball flight tracking.
The minimum width for an adult hitter is 12', but 14' is optimal and is becoming more common every year.
Your minimum length can be 20' or less, as long as you're only hitting off of a T.
If you're planning on doing any coach toss, you'll want your net to be at least 40'.
If you're planning on using a pitching machine, you'll want your net to be at least 55'. You can read more about net size here.
More Batting Cage Netting Frequently Asked Questions