How can I make a sugar free strawberry syrup?
Hmm, you might omit the sugar, reduce the lemon juice (maybe by half or so), and blend the strawberries very thoroughly with the remaining lemon juice, and that way you will get a nice flavorful strawberry sauce.

If, you then strain the sauce very well, and reduce the (basically) strawberry juice to the desired thickness (I'm picturing a honey-like consistency), you will end up with a translucent, thick, and smooth strawberry not-syrup - since syrup is defined by the added sugar, as others mentioned, and assuming the natural sugars in the strawberries don't end up counting enough to make it a "real" syrup.  I don't think it will reach a really syrup-like stage without that straining, since the berry fibers will turn it into a paste instead when the water is too low for a sauce.

This strawberry not-syrup would be much more strongly strawberry flavored, for the same consistency, since the extra sugar needed to make it syrup-thick is substituted with extra strawberry juice, minus water.  You will also get much less syrup out of it, between the volume of the missing sugar, the missing half of the lemon juice, and the loss of the extra water needed to make the final product thick enough.  But, it should act, and store, much like a strawberry syrup would.