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So I was looking at a travel site where when you make a booking, it asks for both your country of citizenship AND country of passport.

I have dual citizenship, and thus am eligible for both passports. So I could I suppose put South Africa for the citizenship and New Zealand for the passport field if I'm feeling like it, but it seems pointless to have the two fields like that.

Then I wondered - is it possible to get a passport without having citizenship of that country?

In Latvia there is a category of people called "noncitizens" - those people lived in Latvia when it got independence, but they weren't Latvian origin so they did not get Latvian citizenship automatically, and got this status instead. Those people use a passport (it is indeed a true passport) issued by Latvia, but they are not considered citizens of Latvia. This also affects the list of countries they could enter visa-free - for example, they're not allowed to US under ESTA, but they can enter Russia without visa if born before 1992 (Latvian citizens need visa).

Also in the USA if you're a permanent resident, you can obtain the "travel document". It is not a passport (unlike the document mentioned above), but could be used for traveling. This document would be issued by the USA, but for someone who is not US citizen or national.

This is an odd example, but nationality law is full of odd cases.

All US Citizens are considered US Nationals, but it is possible to be a US National without being a US Citizen. This situation applies to some people from American Samoa (and Swains Island, with its population of 17 coconut harvesters), residents of the Northern Mariana Islands who have elected to be non-citizen nationals, and children of existing non-citizen nationals who pass on their nationality (subject to special requirements).

Very few people fall into this category (especially as some American Samoans will still obtain citizenship at birth through Jus sanguinis, and others will naturalize), and the State Department didn't want to spend money printing up special paperwork for them, so people who fall into this atypical classification are issued US Passport books that bear the annotation: "THE BEARER IS A UNITED STATES NATIONAL AND NOT A UNITED STATES CITIZEN."

As the Department has received few requests, there is no justification for the creation of a non-citizen national certificate. Designing a separate document that includes anti-fraud mechanisms was seen as an inefficient expenditure of resources. Therefore, the Department determined that those who would be eligible to apply for such a certificate may instead apply for a United States passport that would delineate and certify their status as a national but not a citizen of the United States.

This status occasionally causes problems, as non-citizen nationals may not be able to hold government jobs, vote, or run for office, at least not without obtaining citizenship, which requires the fee, test, background check, etc...

Yes. For example, people born in American Samoa are not US Citizens under jus soli (although they would be citizens if a parent was a US Citizen), but they would travel on a USA passport. This anomaly is a leftover of American colonialism. American Samoans are US nationals but not citizens. Unlike, say, someone from Puerto Rico, even if resident in one of the 50 states, they would not be eligible to vote. (This situation once held for Puerto Rico and the other territories taken in the Spanish-American War.) Source

I suspect other countries have similar situations.

There are a few odd cases, some of which have been mentioned, but I think the main use case for that are actually refugee travel documents (based on the 1951 convention or otherwise) and 1954 convention travel documents for stateless people. Technically they are sometimes called “travel documents” instead of “passport” but they look like passports, are used like passports and you would need to fill in this form if you had one.

Now, by definition, stateless people don't have a citizenship and refugees cannot avail themselves of the protection of their country of citizenship (and in particular they cannot apply for a passport, even if they are still citizens) so in both cases the travel document is issued by the state where they reside, which is not their country of citizenship.

The remnants of British colonialism left behind a number of complex citizenship situations, including special types of passports issued to non-citizens, such as the British National (Overseas) passport.

Wikimedia Commons: Hoising (Note the lack of the words "European Union" on the cover, as such passports do not grant EU citizenship rights)

These passports do not grant the right of abode in the UK. While millions of these passports were issued after the handover, far fewer are in use today, as cheaper Hong Kong SAR passports are often more advantageous.

According to this Freedom of Information request, there are also a number of passports outstanding for British Overseas Territories Citizens, British Overseas Citizens, British Subjects, and British Protected Persons. All hold some form of British nationality, but not British citizenship, and special rules apply to where and how these passports are accepted.

Such situations could lead one to complete your booking form by specifying the UK as their country of passport, but another country as their country of citizenship.

In 1979, six American diplomats were sheltered by Canadian officials after escaping the storming of the US Embassy in Tehran, as dramatized in the film Argo. The effort to sneak them out of the country was coordinated by the CIA and the Canadian government and required Canadian passports so the Americans could pose as part of a Canadian film crew.

Canada's Cabinet met in secret and approved an Order in Council authorizing Canadian passports (under fictitious names) to be issued to the six Americans, as Canadian law would ordinarily prohibit such documents. The diplomats were able to board a flight out of Iran and were reunited with their families.

So yes, it is possible, but this particular route (or any other where a country's government authorizes the special issuance of a passport contrary to normal laws) will require extraordinary circumstances indeed.

When I was working in a hostel in Spain, a few times I checked in people with passports from where they were living instead of from their country of citizenship. There was always something like "non-citizen" printed on it.

For example, Travel Documents near the bottom of http://www.dha.gov.za/index.php/travel-documents2