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.
There are a few reasons that might merit rejection by the UN. 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.
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.
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).
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 piece loving.