Why is NaCl3 possible?
Yes NaCl3 is possible.  

Think of the more common iodine analog.

$\ce{KI + I2 -&gt; KI3}$

For a comprehensive analysis see:

Unexpected stable stoichiometries of sodium chlorides


  At 25-48 GPa, NaCl3 is stable in the Pnma structure, which has 4 formula units in the
  unit cell. Unlike all the other new phases predicted here (which are metallic), this phase is
  a semiconductor. Its structure (Fig. 2(B)) contains almost linear asymmetric Cl3 groups.
  Bader analysis (19) shows that the middle atom in the Cl3-group is nearly neutral, with
  most negative charge on the side atoms (Table 1) and the total charge of this anion group
  being ~-0.8. This result is reminiscent of well-known triiodides and azides with I3
  - and
  N3
  - ions, and of recent calculations on hypothetical H3
  - ions (20), which were found to
  have charge configuration [H-0.81H+0.72H-0.81]-0.9. Pnma-NaCl3 still can be viewed as an
  octet compound, with covalent bonding within [Cl3]--groups and ionic bonding between
  Na+ and [Cl3]-. At 48 GPa, with the formation of a new phase of NaCl3, the octet rule
  breaks down.


This research group compressed chlorine gas between NaCl plates, while heating with lasers and observed formation of $\ce{NaCl3}$.