Updated 1/7/19: I no longer recommend using the method below, and I may not provide any further updates. I now recommend Subadub for studying Japanese from Netflix subtitles. If you want to batch-download subtitles in any language, I recommend Netflix Subtitle Downloader.
I just want the subs!
I have already downloaded Japanese subtitles from over 100 shows and movies, including many native Japanese Netflix originals, and you can grab them all here.
I want to do it myself!
This process involves installing a home theater application called Kodi, and then installing a plugin for it that will allow you to sign into Netflix and then automatically save SRT subtitles for any files videos that you try to play.
The NetflixSubs plugin and installation instructions have been moved to a GitHub page here: https://github.com/Zarxrax/NetflixSubs
Just click through to that page and scroll down for instructions on how to download and install the plugin.
Here is a video tutorial by Matt VS Japan, which shows how to install Kodi and the NetflixSubs plugin, and download subtitles. This uses an older version of the plugin, and some things have changed, but this still gives you a basic overview of how to get things set up.