Besides all the talk about the "Philosopher's Stone". Can anybody or anything become an alchemist? Not including homunculus.
TL;DR Yes, every person can be alchemist, with few exceptions.
Now, to answer this question fully, lets look at what alchemy is.
Alchemy relies on concept of world interconnectivity. World consists of numerous elements, most of which have some connections to other elements, forming interconnected systems, and those systems are interconnected as well, etc. Any action towards one object will also impact all objects connected to it; those objects will impact their relative objects etc. This was explained in episode 12 of Brotherhood series, e.g. food chain is example of one of such interconnected subsystems.
This is source of main principle behind transmutations - you can indirectly affect something in this world, if you know how it participates in its subsystem.
This also explains why you need to learn alchemy, and why most alchemists have limited area of powers. To know how to do transmutation, you need to learn about corresponding systems of world.
Now, important thing - human is part of this system as well. And by changing himself, human changes world around him.
"Fullmetal Alchemist" hyperbolizes this idea: alchemy is basically process of relaying some abstract changes inside alchemist into his surroundings, using interconnectivity of world.
So, basically, since people are part of the world, every single of them can be alchemist. But to do so, you need to learn how world works (or at least how certain areas of world structure work). This is kind of same to scientists - everyone can learn quantum physics, but it is hard, and not everyone has knack for it.
But there is also such thing as Gates of Truth. According to all notes above, it makes sense to assume, that Gates are representing connection of person to world. Therefore, if there is no Gate, person is not connected to world, and cannot use alchemy.