MakeMKV and other DVD ripping software does not preserve EIA-608 closed captions from DVDs. This is the process I use to archive DVD video to include 608 closed captions. You can also upload these captions to YouTube.

If the disc has DRM use makemkv to rip the DVD to an iso file.

If the disc has no DRM, you can use ffmpeg to extract a specific title from the disc:

ffmpeg -f dvdvideo -title 0 -i [path to dvd drive] -map 0 -c copy title0.ts

On windows replace [path to dvd drive] with the drive letter, such as “E:\”
If the disc has multiple titles, open the disc in VLC then go to the Playback → Title menu to view all titles on the disc.

Your resulting .ts file will include the 608 captions. These will play in VLC and some video players, but not all of them. If that's all you want you can stop here.

The native file format of 608 closed captions is a Scenarist Closed Caption (.scc) file. This file contains the raw data stream exactly as it is in the original video.

To extract a .scc file from your ripped .ts file:

ffmpeg -f lavfi -i movie="title0.ts[out+subcc]" -map 0:1 -c:s copy "title0.scc"

If the source video is interlaced, I recommend deinterlacing it for smoother playback. If you plan to upload this video to YouTube you should also upscale it so YouTube allocates higher bitrate your video.

De-interlace and encode to H264, retaining original video size:

fmpeg -i "title0.ts" -c:v libx264 -crf 18 -filter:v "bwdif" -c:a aac -b:a 384k -movflags +faststart "output.mp4"

De-interlace, encode to H264, and upscale to 1080p

fmpeg -i "title0.ts" -c:v libx264 -crf 18 -filter:v "bwdif,scale=-1:1080:flags=lanczos" -c:a aac -b:a 384k -movflags +faststart "output.mp4"

Upload the upscaled .mp4 file to YouTube. Once done, click the “Languages” tab on the video details page.
Click “Upload Manual”, select “With timing” and select your .scc file.