If we're not careful, we can all fall into the trap of taking one verse, making it stand alone, and assigning meaning to it. It is the Holy Scriptures that give us and life meaning, not the other way around. When reading scripture, context is everything. The funny thing about Biblical context is that it is never set in present day western society, and it is never solely about us. It might be for us, but if we want to understand God and His Word, then we must know the historical and literary context.
Paul wrote this letter to a church he founded in Philippi from jail while awaiting trial with execution looming over his head. In the verses leading up to 4:13, Paul says, "I have learned to be content whatever the circumstances. I know what it is to be in need, and I know what it is to have plenty. I have learned the secret of being content in any and every situation, whether well fed or hungry, whether living in plenty or in want." He then says, "I can do all things..."
The "all things" Paul is referring to is really about being content. Contentment is being satisfied with your life no matter your circumstances. Its focus is a state of mind, not a physical ability. It is wanting and being grateful for what you have. The trap of this world is to want more and/or think your life would be better if only... Paul is able to be content in jail, where he doesn't have the freedom, people, and things he is accustomed to. He knows that knowing Jesus is the most important thing in the world, and that is enough for Him. Knowing he is loved and forgiven by the God of the Universe has allowed him to be free from the discontentment and restlessness that plagues so many of us. He is able to be grateful for God's amazing grace and mercy. When we focus on God and His goodness, He can give us the power to have joy and peace of mind in every situation, even if the world is falling apart around us.