You could store files in a Backblaze B2 bucket and serve them via Cloudflare with zero bandwidth fees. Or try DigitalOcean's Spaces offering with their built-in CDN at $0.01/GB for bandwidth.
Hardware video encoding could be done using cheap Hetzner servers that include GTX 1080 GPU's (lookup NVENC) or try Intel Quick Sync encoding (perhaps using OVH's overclocked i7-7700K CPU's).
From Cloudflare's TOS [0]:
>2.8 Limitation on Non-HTML Caching
>The Service is offered primarily as a platform to cache and serve web pages and websites. Unless explicitly included as a part of a Paid Service purchased by you, you agree to use the Service solely for the purpose of serving web pages as viewed through a web browser or other application and the Hypertext Markup Language (HTML) protocol or other equivalent technology. Use of the Service for the storage or caching of video (unless purchased separately as a Paid Service) or a disproportionate percentage of pictures, audio files, or other non-HTML content, is prohibited.
Of course Cloudflare offers commercial plans that feature video CDN, but it will cost you and it will cost you similarly to other video CDN providers.
> disproportionate percentage of pictures, audio files, or other non-HTML content, is prohibited.
If your video sharing app has any decent usage your ratios are gonna get in their red zone pretty quick...
E: typo