The biggest issue would be not being recognized as a country. There are a few basic qualifications that a country must meet to be considered a country in international law. The Montevideo Convention is important here:
The state as a person of international law should possess the
following qualifications:
a. a permanent population;
b. a defined territory;
c. government; and
d. capacity to enter into relations with the other states.
These are very common reasons for rejecting the notion that a country even exists at all (and thus, its UN membership). If your 10-person country lacks a permanent population, or it doesn't have any ability to trade, for instance, with other states, or it doesn't have a proper government, it would easily be rejected on these grounds.
You'd also have to convince them that there was no other legitimate government of your island. Just owning it might not be enough, since it might still be under some other nation's jurisdiction.
There are a few reasons that might merit rejection by the UN even if they accept that you're a country. Particularly, quoting Chapter II:
Membership in the United Nations is open to all other peace-loving
states which accept the obligations contained in the present Charter
and, in the judgment of the Organization, are able and willing to
carry out these obligations.
Can your 10-person country really carry out the various obligations listed in the charter? What if those obligations require more than 10 full-time positions, for instance? While these rules are not applied consistently (see below), they could and likely would be applied against your hypothetical country.
That said, international recognition in general, including UN membership, is a highly political process, dependent less on fairness and objective criteria than on international relations.
For instance, Palestine (and to a lesser extent Israel) is recognized based less on objective assessments of its status as a country, and more on how much a country values its relationship with Israel and Israel's allies. Taiwan, for instance, does not recognize Palestine, likely because of its strong ties to the US. Conversely, Cuba and North Korea do not recognize Israel, despite their apparent dissimilarity and lack of geographical proximity. This likely owes something to both states (at least until recently in the case of Cuba) already being isolated by the US and its allies. Similarly, the majority of states that don't recognize Israel are Arab or Muslim states, which likely benefit more from trade with each other than with Israel, and who might face strong domestic pressure to adopt a tough stance on Israel.
Another instance is that of South Africa, where at one point it faced expulsion (for reasons that could also, in theory, serve as grounds for not admitting a country) due to its system of apartheid. This motion failed due to the vetoes of the US, France, and Great Britain, less because of philosophical support for apartheid or fear that they could be expelled (though those might have played a role), and more because South Africa was an important trading partner.
The most extraordinary example is Taiwan itself, once not only a UN member state but a permanent member of the Security Council from its founding. Due to political considerations, in 1971 it was determined that the PRC was the legitimate representative of China, and Taiwan no longer has its membership, and has lost a great deal of international recognition (including, for instance, by the US). Essentially, trade and normalization of relations with China became more important than "opposing Communism" to the US and its allies, and it switched its recognition accordingly.
The question of what constitutes a peace-loving state is also applied rather loosely. To my knowledge, no state has been rejected for being to violent, though one suspects, for instance, that an application by ISIS might be rejected on such grounds. However, states that engage in frequent war with at least one UN member state (the US, Iraq, etc.) have member status. States with violent dictators (North Korea, for instance) have also not been rejected on the grounds of not being peace loving. In fact, even Syria (whose leader has killed a great number of its citizens) is still a member.
Given this relatively arbitrary nature, it should be pretty easy to see that many countries, especially the Security Council members, would choose not to enter into relations with small, unprofitable partners that might cause a big headache down the road, including the example country.
