When you say it doesn't start it just appears dead? Were there power indicator lights? Are they on even if the drive doesn't start? If it's an older spinning hard drive and is not making any sound or vibrations then your options are limited. You can try checking all the cable connections, data and power ( if there is external power brick test that). Also try it in another computer. It could be the power side of it is broken and the drive is fine.
If the problem is the hard drive itself you would really need to send it to a data recovery company. There are a bunch of them but the larger companies can be ridiculously expensive. Thousands of dollars. Honest companies should give free diagnosis BEFORE they give you a quote and a have a no data recovered no cost to you guarantee.
One company that seems to be honest and well reviewed is ...
$300 Data Recovery: Hard Drive, RAID, SSD & Flash Repair in Los Angeles
Call them and see what they can do for you.
Also there is this company which is one of the larger ones but has good reviews and might be worth talking to.
ACE Data Recovery | Hard Drives | RAID | SSD | Apple Mac
Disclaimer: I know nothing about either of these companies and cannot recommend them from personal experience. I found positive reviews for both. They are knowledgeable and in the business of data recovery and are at least worth talking with.
Good luck.