Splitting a 240V line into two 120V lines
Yes, you can certainly do this. It is referred to as a Multi-Wire Branch Circuit. The two hot legs share the neutral. It is quite common.

Note: Make sure you pigtail the neutral and ground wires, do NOT daisy chain them through the receptacles.

Here is a pertinent National Electrical Code reference:


  210.4 Multiwire Branch Circuits.
  
  (A) General.  Branch circuits recognized by this article shall be permitted as multiwire circuits. A multiwire circuit shall be permitted to be considered as multiple circuits. All conductors of a multiwire branch circuit shall originate from the same panelboard or similar distribution equipment.
  
  Informational Note No. 2: See 300.13(B) for continuity of grounded conductors on multiwire circuits.
  
  (B) Disconnecting Means.  Each multiwire branch circuit shall be provided with a means that will simultaneously disconnect all ungrounded conductors at the point where the branch circuit originates.
  
  Informational Note: See 240.15(B) for information on the use of single-pole circuit breakers as the disconnecting means.
  
  (C) Line-to-Neutral Loads.  Multiwire branch circuits shall supply only line-to-neutral loads.
  
  Exception No. 1: A multiwire branch circuit that supplies only one utilization equipment.
  
  Exception No. 2: Where all ungrounded conductors of the multiwire branch circuit are opened simultaneously by the branch-circuit overcurrent device.


Good luck!