Is it possible to hand start modern planes?
You sure can!

In reality the engines in modern prop planes are not all that different (in some cases even identical) to the engines of the hand crank days. The reason many older planes were hand crank was simply to save the weight toll of a starter motor and battery to drive the system. Some older aircraft have no active electronics (the piper cub comes to mind). Remember the engines magnetos drive the spark for the ignition system unlike in a car where the spark is driven by the electronic system. As long as proper precaution is taken its more than possible. The FAA even explicitly discusses it in their airplane flying handbook (p 2-13) 


  A spinning propeller can be lethal should it strike someone.
  Historically, when aircraft lacked electrical systems, it was
  necessary to “hand prop” an aircraft for starting. Hand propping an
  aircraft is a hazardous procedure when done perfectly. The
  consequences of not mitigating the hazards associated with hand
  propping can lead to serious injury, fatalities, and runaway
  airplanes. All alternatives must be considered prior to hand propping
  an aircraft and, when a decision is made to do so, the procedure must
  be carried out only by competent persons who have been trained to
  accomplish the procedure, understand how to mitigate the hazards, and
  take all the necessary precautions.


They also provide a pretty clear checklist on how to do it. 


  The procedure and commands for hand propping are: 
  
  • Person out front
  says, “GAS ON, SWITCH OFF, THROTTLE CLOSED, BRAKES SET.” 2-14 
  
  • Pilot seat occupant, after making sure the fuel is ON, mixture is RICH,
  magneto switch is OFF, throttle is CLOSED, and brakes are SET, says,
  “GAS ON, SWITCH OFF, THROTTLE CLOSED, BRAKES SET.” 
  
  • Person out front, after pulling the propeller through to prime the engine says, “BRAKES
  AND CONTACT.” 
  
  • Pilot seat occupant checks the brakes SET and turns
  the magnetos switch ON, then says, “BRAKES AND CONTACT.”


As far as i know, you can not hand start a turbo prop (which some GA planes have) due to the higher RPM needed for start.