Why can't electrons accumulate at a point in a circuit? The most common reason we get for why the same current flows through all resistors in series is that because electrons cannot accumulate at any point. But why is that? Why can't some free electrons move from their free state to positive ions and similarly why can't new valence electrons join free electrons? Would it not change the current flowing in the circuit? Can someone clarify what I am missing?
 A: The reason is quite simple: Elecrons repel each other, and they do so quite strongly. The number of electrons is bound to be approximately equal to the number of protons in the solid (out of which only the free electrons can move).
A fun exercise is to calculate the coulomb force resulting from displacement of all electrons in a penny by one meter. 
A: This question is a bit confusing. Electrons CAN accumulate in a point in a circuit, that's how capacitors charge up.
And the second part of your question concerning free electrons and ions is not very clear.
A: With real wires, there will be some parasitic capacitance between any given wire and whatever other wires are nearby. Because of this some charge will accumulate in the wire (and an equal charge will be repelled from some other conductor, leading to no net charge accumulation in the system as a whole). However often (and if we design carefully) this capacitance is small enough that we can ignore it and get a "good enough" model to predict how our circuit will behave.
In comments you asked,

IF THEY CAN THEN HOW WE CAN OBTAIN A STEADY CURRENT THEN?

Because once charge has accumulated in the wire, it will oppose further accumulation. We eventually end up with a steady state charge (attributed to the parasitic capacitance) and a steady state current through the wire, which doesn't depend (at all in many cases) on the accumulated charge on the wire.
A: There are multiple quantum mechanical ideas involved:


*

*Due to the interaction between the charged particles, the solid obtains a band structure. As you already stated some electron occupy states "close to the ion". These electrons are bound to the ions and can not travel freely inside the conductor. In contrast, some electrons form the so called free electron gas inside the conductor. They are the charge carriers (i.e. current carrier).

*The band structure is also due to Pauli's exclusion principle (which is a symmetry principle and not an interaction). Pauli's principle states that two electrons can't occupy the same state (simplified version). Thus, if a bound state ("near an ion") is occupied, no electron from the free electron gas is allowed to occupy this state. The state is filled.


Now the idea of a solid is to fill up the allowed electron states starting with the states which posses the lowest energy. These are the bound states (they are "closer" to  the positively charged ions). If the solid possesses only "few" electrons, such that all of them find a bound state, then we call the solid a non-conductor. In contrast, if the solid possesses "many" electrons then some of the electrons must occupy free electron states and we call the solid a conductor (I won't go into semi-conductors, here) -- note that I define "few" and "many" with respect to the available number of bound states. 
I believe that the simplest picture is that the "bounded electrons" shield the ions from the "free electrons". Just as every single atom can occupy only a certain amount of electrons before it is electrical neutral, this is also true for a solid. However, since two ions can share a "bounded electron", they don't need all their electrons to compensate their electrically positiv charge. 
