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The OSU Open Source Lab gives machines to groups in their datacenter: https://osuosl.org/services/hosting/

It has hosted quite a few famous services.

Which famous services?

I doubt OSU is going to host F-Droid. It doesn't even sound like F-Droid would want them to host it.

https://osuosl.org/blog/osl-moving-to-state-data-center/ mentions several major, famous services/projects that OSUOSL either has hosted in the past or is still hosting: kernel.org, Debian, Gentoo, Drupal, OpenWRT, OSGEO. https://osuosl.org/blog/osl-future/ also mentions hosting Mozilla at the time of the Firefox 1.0 release, and having previously hosted Apache Software Foundation. Closer in relevance to F-Droid, OSUOSL hosts the GitLab instance used by postmarketOS: https://postmarketos.org/blog/2024/10/14/gitlab-migration/

F-Droid is the best known non-corporate Android App Store... Why wouldn't they be willing to host it?

It's a critical load-bearing component of FOSS on Android.