Far as I know, the major calendaring protocols all use HTTP as it is. Do you just want a mediator/self hosted backup? Or do you want to host your own calendars?

If it's the latter, Radicale is awesome.[1]

1: https://radicale.org/3.0.html

I think my biggest difficulty is that I'd like to be able to look at a duration of time and determine the events within that range.

Pulling raw iCal data over HTTP is possible but if you say, set up a recurring event several months back, you need to fetch that original event (even if it recurs every day) to include it in your calculations... I'd love a self-hosted version of this Nylas API [0] for example.

[0]: https://docs.nylas.com/reference#calendars-free-busy

We’ve working on this for one of our side projects. When you have hundreds of calendars and need to find availability for some of them on a timeslot, or fetching the slots for a period of time (I.e, a week) it becames hard really fast.

We are evaluating if it may be interest on it to release as a separate SaaS.

You can find my email on my profile if you want to discuss further your needs.

iCalendar files over HTTPS via OAuth?

https://tools.ietf.org/html/rfc5545#section-1

CalDAV is already built on HTTP, it's just implemented with a lot of gotchas and ancient history: https://tools.ietf.org/html/rfc4791#section-1

http://www1.udel.edu/CIS/software/dist/google/calendar/java.... for Google Calendar ics feed documentation (there might be better links, this just came up first)