Why? What things need to happen every exact 7 days? Isn't it rather the case that we do some things weekly because that is a conventional unit of time?

And if you are designing something so that some process needs to be performed on it every week, you should be building some tolerance into that time-period. Which the eight day span, once every year, would test.

I think we would need to be more precise about our measures of lifetime for some things (recent e.g. would be MH370's black box battery life rating) but the idea of measure lifetime with days since an event is already used quite widely (I rarely heard the phrase 2 weeks since that event) and could be adjusted to without much difficulty.

Financial and similar institutions would likely treat the festival day(s) as they do any long weekend at the moment. Pay cycles could be adjusted with end of year 'bonus' to cover out of cycle days.

I'm sure there's something but we've dealt with quirky calendars for a long time so really we're just attacking the same problem that we currently have with 30 day fixed cycles.

Once we get off the surface of the earth, seconds are more useful than days. Or, more likely, Ksecs and Msecs. A kilo-second is 16.67 minutes, 100 Ksecs is 27.7 hours, an Msec is 11.57 days. 30 Msecs is 95% of an Earth year. These are all useful measures for humans to work in.