To answer the OP, if you were to run a new 14/2 (15 amp) circuit it will be adequate to handle a washing machine. If it is in the basement or within 6' of a sink it should have GFCI protection. However it will not be in compliance with the National Electric Code as it would be required to be a 20 amp circuit. Again it will handle the load and be perfectly safe.
Because you have old 2 wire (no ground) non metalic sheathed cable (romex) all of your outlets will show open ground. (of course it is open - there isn't a ground) Per NEC you may replace the older 2 wire receptacles with 3 wire receptacles provided that you install GFCI protection upstream and label the 3 wired receptacles alerting users that there is no ground.
You should check permitting requirements in your area to see if you are allowed to do the work as an unlicensed homeowner.
At a minimum I would install new circuits to all the major appliances. Refrigerator, dish washer, range, dryer, etc.
The original wiring is not 16 gauge. The older 14 gauge is just a lot softer and may seem to be smaller than today's.
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