For a small private community you can increase the max upload size and upload those to Discourse.
There are some things to consider:
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videos consume a lot of space so you will want to setup some form of block storage (s3 and the clones) and CDN
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if you are putting those in a s3-like bucket you could even bypass uploading those to Discourse and just put em on the bucket and oneboxing
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Be careful on the codecs to ensure your video can be played everywhere
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You said the videos need to be private. Uploading videos directly will allow people to right click and download the original. Not sure if that will be a problem in your application.