I haven't used OBS in some time but one thing I can give you some help with is Handbrake.
In handbrake when you open a video file you're presented with a "Summary" of the video. On this tab it will include information about your video like the file format and the resolution of the video.
If you make your way over to the "Video" tab (4th one along) you can edit all the settings for your video here and this is where you will reduce the size of your video.
On the "Video" tab the preset is default at "Fast 1080p30" which means a resolution of 1920 x 1080 for the video, and a frame rate of 30 frames per second. Now depending on what resolution and what FPS you recorded your video at in OBS you may not need to change these values, but for the sake of this answer I'm using a test video shot at 1920 x 1080, 60 frames per second, so I will go ahead and change the "Framerate (FPS)" option to '60'.
After changing the resolution or FPS, to the right hand side of those options is a "Quality" setting. This slider is what you need to play around with to get the file size down. You mentioned you didn't care about video quality so I would use a value of around 35 to begin with and then render your video and preview to see if the quality is somewhat view-able and the file size isn't too large.
After you've made changes to the settings you need to click "Browse" in the bottom right and find somewhere for your rendered video to be saved, and give it a file name as well.
Once chosen you can go ahead and click "Start Encode" at the top and Handbrake will begin to encode your video but with the selected settings. Once done you can go ahead and view the video and file size to see if it's what you're after. If it's not just adjust the quality setting some more until you find something you're happy with.
Just an example of what Handbrake is capable of:
I started with a video shot at 1920 x 1080, 60FPS. It was 35MB in size and I ran it through Handbrake with the "Video Codec" the same (H.264), and the "Framerate" I changed to 60 (as it defaults to 30 on the selected preset). I set the quality to 35 and encoded the video. The file size is now 3.7MB! And the quality is still bearable!
I have came across an issue in the past where OBS can record at a variable framerate meaning the audio didn't match with the point in time of the video. If you ever find this is the case then there is a setting in the "Video" tab under the "FPS" option in Handbrake that fixes this Tick the "Constant Framerate" box along with all the other settings and it should make your video re-align it's audio and video channels and use the same FPS for them both, eliminating the problem!
Hope this helps!