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.

You can make your strawberry sauce syrup without sugar easily just by replacing the sugar with stevia and then adding it to a proper amount of gelatin water for the proper consistency. It won't be the same for sure but it will be almost sugar free and thicker than water but not have the exact consistency of a syrup. Maybe a little xanthan gum along with the plain gelatin would be a little closer though.

You could replace the sugar with some artificial sweetener. I don't know the exact ratio, but I believe you can use less sweetener to an equivalent amount of sugar. But by definition syrup contains sugar, so you wouldn't be able to eliminate it (or a replacement completely).

I suppose, depending on what you're doing with it (topping for dessert, waffles, pancakes), you could just take the strawberries, add some water and lemon juice, and reduce it and make a sauce instead of a true syrup...

Use Agave Syrup/nectar instead. Reduce the quantity a little, about 2/3cup instead of your 1 cup of sugar.

Since you're using a liquid instead of real sugar, you may need to then reduce the amount of lemon juice. To keep a lemon flavour, you could try infusing the syrup with a slice or two of lemon while it's cooking. Definitely remove any lemon rind before bottling or using as it will be too bitter otherwise.

I always use stevia. It's not an artificial sweetner, it's natural and it comes from a leaf. You can buy the extract which is super super sweet (a tiny level scoop it usually comes with equalls a tablespoon) or you can buy stevia sugar which is supposed to come to the same volume as sugar (so you don't have to add bulk) but it's more expensive. I usually buy the liquid stevia drops since I make a lot of chocolate at home and it just mixes right into other liquids I'm using. Nice. Liquid stevia can even come in different flavours. Oh, right, and it doesn't have any carbs/calories!