Your question is unclear:
it has to be assumed that there are more than zero domain names in existence which have not been purchased yet. Therefore, does that mean that they aren’t owned?
A domain either exists (is registered) or not. If it exists then it was registered (and paid for, except in rare exceptions) by someone that decided who is its current owner.
If a domain doesn’t exist, then obviously it is not owned by anyone. Some domains do not exist and are protected in such sense they can never exist, the registry would deny any attempt to do so because they are in some reserved list of names, or are covered by various “blocks”. You could say, with a stretch, that in those cases the domains are owned by the registry but in reality the important point is that they don’t exist at all.
If so, would I be able to theoretically “acquire” said domain names without purchasing them
Some TLDs provide domain names for free, like
.tk. But outside this known rare case, all domains are to be paid for to enjoy using them. As such, you can not acquire a domain name without paying for it.
considering that they aren’t anyone’s intellectual property?
Not sure what you mean by that as there is obvious overlap between domain names and things like trademarks, so all previous rules and processes distinctly put priority on trademarks/protected or registered names, over domain names, as long as the domain name is not registered before said trademark.
You are free to choose any currently non registered and available domain name to register it by paying whatever amount the registrar you chose is billing you for, as long as you follow the eligibility requirement of the given TLD you are registering in (an important information missing from your question, rules change per TLD around who is eligible or not to register names), and you are aware that you may infringe others rights, especially if you try to register known trademarks, in which case you can expect to have further interactions like a lawyer’s letter, or procedures like UDRP or URS (in gTLDs as mandated by ICANN and some ccTLDs as is or with their own form of disputes resolution mechanisms that are often close in spirit to those ones) thrown at you, at which point you have to decide if you want to keep the name and defend it because you think you have merit, or you delete it or you transfer ownership to the asking party.
But, more generally, it is difficult, for any given arbitrary name, to really know, worldwide, if it is covered by someone’s intellectual property claim. There are obvious cases, like Apple in Rob’s answer, where any defendant would be hard-pressed to try opposing it didn’t know about this trademark, but then there a lot of gray areas.