Fermionic (or Bosonic) state vs Entangled state One can see that the wavefunction for a system of two electrons (not very far apart) is one that cannot be written as a tensor product of individual states.
The same is true for a bosonic state.
For instance, production of an electron-positron pair has a zero total spin, the state of which looks identical to an entangled system of two electrons.
Is it possible to have entanglement between indistinguishable particles?
Does one have to first make them distinguishable like taking them far apart and then try to entangle them?
The question arises from the fact that the fermionic state looks like an entangled state. And I don't understand whether this is something that is just a coincidence or is there something else to derive from this.
 A: 
Is it possible to have entanglement between indistinguishable particles?

Yes.

Does one have to first make them distinguishable like taking them far apart and then try to entangle them?

No.
An instructive (counter) example is parametric down-conversion, which is commonly used to create entangled pairs of photons:
A single photon with spin up or down is send through a crystal (glass), where it is converted into a pair of two photons with the same spin but slightly different directions. Those can be separated and then used as an entangled pair. Since the spin of the initial photon is unknown, the wavefunction after down-conversion is:
$$|\psi\rangle = \frac{1}{\sqrt{2}}( |\uparrow\rangle \otimes |\uparrow\rangle + |\downarrow\rangle \otimes |\downarrow\rangle). $$
The photons are still "distinguishable" by their position, although nothing would be changed if you swapped them and the wave-function is symmetric. And they are entangled even before you can distinguish them, so they do not have to be separated to be entangled. In fact, you can only create entangled pairs of particles from a common origin (they have to be close in the beginning).
