While Mr. Klocking implies in his answer that fallibilism (IEP) easily and consensually among professional philosophers overcomes the problem of skepticism, contemporary skepticism (IEP) lives on, and that's okay. I'm going to outline a broader line of attack by a contemporary philosopher and go above and beyond a reference to a book. I'll refer to Robert Audi's Epstemology and it's chapter "Skepticism" as a basis for the response.
Robert Audi's approach to answering moderate skepticism is as follows:
To show a belief is tenable, first show in a second order claim that the belief in belief is possible with a general premise:
(1) An attentive belief to the effect that one is now in an occurrent mental state, such as thinking, is justified.
where "attentive belief is on based on careful attention to the matter in question, and where the justification is not absolute but simply strong enough to make it appropriate for a rational person to hold the belief".
(2) I have an attentive belief I am now in such a state, namely thinking.
Therefore, by deduction:
(3) My belief that I am thinking is justified.
He goes on to admit this reason is defeasible, such as applying a proposition about a history of recent troubles with hallucinations.
So, in short order, the notion that skepticism is indefeasbile, is not only open up to criticism through showing that reason itself, and hence the very conclusion is subject to defeasible conditions, but also by providing a positive argument that justification of belief is one of degree, not black and white, which would be an obvious false dilemma. Rationality entails accepting the fallibilistic nature of knowledge, thus dissolving the question of is there or isn't there knowledge.
So, one can accept both skepticism and credulity admitting they lead to fallibilism, by recognizing, as Audi puts it:
"These questions produce a tension. I want to believe [a certain proposition]... But I also want to avoid believing [the proposition] if it is not [true], for I have a deep-seated desire to avoid believing falsehoods... the former inclines us to believe readily... The latter ideal pushes us toward a kind of skepticism."
Thus, not only is the skeptical problem overcome by recognizing that radical skepticism itself is subject to skepticism, but positively it is the actual state of affairs of all people to find a balance between credulity and skepticism every day of their lives because rational people make attentive and reasonable presumptions pragmatically. To appeal to the classical logic of Agrippa is a good exercise, but accepting a non-naive realism is hardly a problem in contemporary discourse because of the acceptance of non-classical logics, a knowledge of the psychological, and a lot of progress made by epistemologists since the time of Pyrrho of Ellis.