Do questions about man-made objects in space (such as Hubble, Voyager, the Apollo shuttles, etc) fit into this site, or is the site more aimed at cosmology and astrophysics? (the current questions seem to lean more towards the latter)
If not, is there a StackExchange site which they fit into better?
Stack Exchange has two separate sites that could cater such questions, Astronomy and Space Exploration. On which of the two they would likely fit better is though down to what they're actually asking, sometimes even a preference of their authors where they would rather see them answered, since these space probes and space observatories that you mention have both technical aspects as well as do science that is largely astronomy related.
Some questions will inevitably ask about both aspects though, afteral their mode of operation and science they are doing are inseparable, with Kepler being an excellent example due to one of its reaction wheels non-operational, technically limiting science it is capable of performing. So this remains a bit of a gray area, but scopes of individual Stack Exchange sites will inevitably at least partially intersect one another. Enforcing too strict borders between sites likely wouldn't be productive; askers would avoid coming close to it in fear of their questions being rejected, which would only lead to a too wide gap in scope in reality, even if they should cover all bases in theory.
So this boils down to what we defined as being on scope for Astronomy.SE, which does cross over to what's on scope on other Stack Exchange sites to some extent, mostly with Physics that does accept astronomy questions too, but all of this is covered in many threads here in Astronomy Meta, e.g. How to draw a line between Astrophysics and Physics, if there is any possibility? or a plethora of other threads discussing scope, so I won't go into it on this occasion.
As far as questions that are borderline on-scope or in theory equally befitting also other Stack Exchange sites' scope, the general consensus is that unless they're strictly off-topic where posted, we shouldn't push for migration. It is fine to ask its author about it, present a possibility of migration in the comments, but unless it is requested by their authors themselves, they should stay where posted.
As a rule of a thumb though, if you're asking about spacecraft operation, technical aspects of them, and similar, then that belongs on Space Exploration, and if you're asking about science of space probes and space observatories that is astronomy related, then Astronomy would be the place to ask about it, with planetary-science being one example that is catered for on both sites equally. And if you're crossing both aspects in your question, then it is up to you where you deem you'll get better answers. But please don't cross-post your question on multiple Stack Exchange sites.
I'm new to Astro.SE, but I think Hubble in particular fits in the scope of the site. All Hubble does is make astronomical observations, 24/7. Without the data from Hubble and other observation satellites in orbit many questions here wouldn't even have an answer. All Hubble data is about astronomy. So Hubble is definitely in.
Voyager's mission was more limited, but if we accept questions about the solar system, we have to accept Voyager as well.
Apollo OTOH doesn't belong here in my opinion. The Apollo project focussed solely on the moon, and it was primarily a geopolitical project, rather than astronomical. (Though the "geo" is a misnomer of course.)
