In every society, there will be individuals who are no in a position to look after their own self-interests, such as the elderly, infants, children, and the physically and mentally infirm.
Some might think this requires other to sacrifice their own freedom and apparent self-actualization, which would be a serious ethical problem for Objectivism, because in Objectivism, you are not supposed to sacrifice yourself for the sake of another.
An Objectivist can counter this problem by saying that somebody will take care of these types of people because they value benevolence or charity, but is it true that what an Objectivst cannot say, is that these types of people are owed care by virtue of being human beings?
The examples listed in this question are all united by the concept of "need," put forth as constituting a claim on the lives of others (by way of an alleged "claim" on "society" as a whole). Objectivism does not recognize or accept that kind of principle of "need."
However, Objectivism certainly recognizes and upholds causality, which can arise in regard to chosen (contractual) obligations and also in regard to the obligations of parents for their own children (including their infant children). By including children in the question's list of the allegedly "needy," the question creates a classic "package deal" resulting from the fallacy of definition by non-essentials. The non-essential uniting the "package" is "need." What Objectivism identifies as essential is causality (or its absence), not mere "need."
Objectivism also denies that there is any such thing as "society" apart from the individuals who comprise it. Objectivism starts with individuals and asks what kind of social-political-economic organization is most consonant with the nature of man and existence. Objectivism identifies man as a rational being, with reason as his basic means of survival, who can benefit greatly from living in a society that upholds individualism and individual rights -- and which does not impose unchosen obligations on anyone (beyond the basic precondition of respecting others' individual rights).
[I]n Objectivism, you are not supposed to sacrifice yourself for the sake of another.
Objectivism does not say there is an obligation to refrain from self-sacrifice. Objectivism leaves everyone free to live (or not) as he sees fit, so long as he respects the individual rights of others. If he wants to die, he is free to do so. Objectivism says it would not be "good" for him to do so, because self-sacrifice is anti-life, and man's life qua man in the standard of value that Objectivism identifies as the only objective standard of value for man, the only objective basis for the concept of the "good." But Objectivism does not say to man, "Thou shalt not be non-objective," nor even "Thou shalt strive to remain alive." Objectivism is not a philosophy of ethical "commandments." The choice to live (or not) resides with everyone individually. What Objectivism fights for is the right of everyone to exist for his own sake, neither sacrificing himself to others nor sacrificing others to himself, with freedom from coercion by others.
Update: Charitable Institutions
From a comment:
I don't believe there would be government institutions for such humans under an Objectivist form of government.
Those who advocate "government institutions" to care for the "needy" are advocating the initiation of physical force against others, although they may try not to acknowledge it too openly, and may try to justify it as unavoidable in the nature of "society" -- with individuals viewed as "resources" belonging to the "social whole." There is no reason why charitable institutions need to be created and run by government and paid for through initiation of physical force against those who are capable and productive. For an excellent, highly concise statement of the Objectivist view of charity, refer to that topic in They Ayn Rand Lexicon.
Update: Collectivized Ethics
The same commenter observes further:
Mentally ill humans walking the streets are not directly harming or benefiting me but I don't enjoy seeing this kind of misery. Other than repressing all feelings of kindness and empathy, I am not sure how to solve this.
Ayn Rand answers this type of issue in considerable depth in her article, "Collectivized Ethics," published as Chapter 10 in The Virtue of Selfishness: A New Concept of Egoism. The article begins:
Certain questions, which one frequently hears, are not philosophical queries, but psychological confessions... Objectivists will often hear a question such as: "What will be done about the poor or the handicapped in a free society?"
The article describes this as "a perfect example of how one refuses to accept an adversary's premises as the basis of discussion." The article goes on for nearly six pages explaining the issues and Objectivist answers in depth.
If anyone seeks advice and/or assistance to overcome emotions of doubt and fear at the contemplation of Objectivism's advocacy of rational egoism in ethics, the approach one will need to take is to learn to uphold reason as man's basic means of survival, with initiation of physical force against others as anti-reason and anti-life. One will need to learn consistently to assert reason over emotion in deciding on a course of action, which anyone can do so long as one remains human (qua rational and volitional living entity). Do it often enough and consistently enough over a long enough period of time, with a consciously thought-out, rational understanding of why reason must prevail over emotions as one's basic tool of cognition and decision-making in action -- and one's habitualized emotional resistance will slowly fade. This is not suppression of emotions; one does not try consciously to avoid feeling the emotions. One merely allows contrasting, reinforcing emotions whatever time it takes for them to become one's normal, unconflicted response to one's reliance on reason. (In some cases, this may be assisted greatly by competent professional psychological counseling.) For further explanation of automatization and emotions, refer to those topics in The Ayn Rand Lexicon.
The commenter also alleges that private charities have "not worked well." This is context-dropping. Today's mixed-economy context must never be confused with a system that consistently upholds individual rights and, as a consequence, upholds laissez-faire capitalism and free markets. Government interference with free markets distorts their operation, especially as the controls endlessly grow and stifle free markets ever more severely.
The commenter mentions: "Mentally ill humans walking the streets..." -- and then proclaims: "I don't enjoy seeing this kind of misery." The commenter should ask himself: Whose streets? He doesn't own the streets and has no rightful say in how they should be run. They would all be private property in a free society. He should also ask himself: by what right does he propose to dispose of others' lives and property to "solve" a "problem" on streets he does not own? And, above all, does he really feel strongly enough about the "needy" to help them himself, as best he can, and/or to contribute financial aid to private charities, if he really wants to help?