Can I Drain Varicose Veins with a Syringe?

You cannot really drain the varicose veins in any permanent way because there are even larger veins that are feeding them; this means, you could lose lots of blood and cause yourself serious, life threatening harm with self-treatment.    The best way to treat them is to have them evaluated by ultrasound and diagnose the venous insufficiency (the veins feeding them), and have them treated appropriately, usually via Laser Ablation that can be done under local anesthesia.  Good luck!

Don’t do this at home, or anywhere.  There is nothing to be gained by draining the vein.  It can cause damage, infection, and blood clots.

Don't do what you suggested, i.e. 'draining a vein' as this will cause a blood borne infection. I saw a patient once who heated a needle and stuck it in her veins and developed infective thrombophlebitis and blood borne sepsis with MRSA which destroyed her aortic valve. She required replacement of her aortic valve. She also developed a stroke from the infected bacterial vegetations that embolized to her brain. 

Varicose veins are formed due to damaged valves of the venous system that allow pooling of the blood with resulting enlargement of the vessels.  Unless you address the underlying problem (the damaged vessels), simply "draining" the vessel by aspirating blood from the varicose vein will not achieve any lasting results.  You may cause damages in forms of possible infection, bruising, hematoma, pain, and clot formation (thrombosis) of the vessels.  There are well established and effective treatments for varicose veins, such as Sclerotherapy, endoveous laser treatment, or possible transcutaneous (non invasive) laser treatment if vessels are small enough.  Please do seek for professional help rather than contemplating sticking a needle to your veins.  Best of luck.