Except that the year is based on the solar orbit, which has some quite obvious effects via the axial tilt causing seasons.
The month is based on the lunar orbits, which are a bit more arbitrary now and less important with an urban rather than agricultural economy, but still not completely arbitrary.
Weeks are a bit arbitrary, as far as we can tell, but very widespread and we seem to be stuck with them.
I'm always blown away by people who don't understand weeks.
A lunar month is 28 days.
There are few divisors of 28. 2, 4, 7, and 14 are the only ones. 28 days is too long for convenience if it's your shortest grouping of days. 2 days is too short. So, 4x7 or 7x4. Having 7 4 day weeks every month is probably a bit less convenient than 4 7 day weeks. Hence, 7 day weeks. (Or two 14 day weeks, which I think you essentially used to see some of the time: it seems like 17th-19th Century Britain was more aligned on fortnights than weeks).
Not NECESSARY in the same way that days are, but it's not exactly mysterious how they came about.
Edit: Oh, also, in terms of preferring 7 or 14 to 2 or 4, think about the lunar cycle. It easily breaks down into 7 or 14 day periods (7 days = new to half, or half to full, or full to half, or half to new. 14 days = new to full or full to new. 4 days = ?)
Edit2: Of course, a lunation (full moon to full moon) is 29.5 days, as opposed to the sidereal month (same part of sky to same part of sky) of 27.3 days. But I suspect that in antiquity, most people dealt with a conventional 28 days for both.
Weeks mostly have to do with the market cycle, which has been anywhere from 4 to ten days depending on the local economy. The Chinese in particular used 5 or 10 day weeks for quite some time. That works better for lunation than 7.
The seven day standard seems to have worked as a reasonable compromise across a wide variety of ancient economies and plan
You're not going to convince either the Christians or the Jews to do a week that isn't 7 days long. That number is more or less fixed by the sizeable minority who need it for the rest of their habits.
The solar year is convenient so that one can speak of "the average historical weather for this month" and such... but it's true that many others have done lunar calendars etc.
Muslims use the lunar calendar for religious occasions (Ramada, Hajj, Eid) but use the Christian, solar calendar (like the rest of the world) for everyday life.
The day and year are settled astronomically. As for the week, it's been in use for at least 2500 years continuously without ever skipping a day and now virtually every country uses it. The month, however, has variable length, has been redefined every thousand years or so, and has competing systems from Islam, Hindu, Jewish, Japanese and other cultures. The month can change, but the week never can.
The month is based on the lunar orbits, which are a bit more arbitrary now and less important with an urban rather than agricultural economy, but still not completely arbitrary.
Weeks are a bit arbitrary, as far as we can tell, but very widespread and we seem to be stuck with them.