How does a radioactive nucleus emit electrons when it doesn't contain any?

When a beta decay occurs there are a couple of things you need to remember. First of all you are right, the nucleus doesn't contain any electrons. It does however contain neutrons and protons. We know that protons have a +1 charge while electrons have a -1 charge, these cancel out when they are balanced. Now let's say we take a neutron of zero charge and split it into a +1 and -1 charge so that it is still overall zero charge. We will get a proton and electron, but the electron doesn't want to exist in the nucleus and so it is emitted. This is what happens in a radioactive nucleus in order for it to emit beta radiation.