Is there any way of getting rid of varicose veins and spider veins in the legs without surgery?

Yes. People who sit at desks (or anywhere) for long periods of time develop this. I actually keep my wifes legs smooth by doing deep leg massages about once a week. You can actually feel the lumps in the legs, but after a while they smooth out. The rub starts just below the buttocks and continue to the end of the feet. I usually do just one leg one night and the other the next night because it actually hurts her. Repeating the rub about once a week keeps the legs looking young. It won't happen in one session but over time they will go away. But it has to be maintained indefinitely. By the way, my wife is 40. Outside of massages, being highly active really helps.

Those vein, capliyaries, deliver blood to your legs and other parts of the body. You should not do surgery to have them removed for cosmetic reasons.
Consider regular brush scrubbing in the shower, once a week. And skin darkening lotions, these only come out spring or summer. Instant sun tan.
once you have them they are there to stay....rec. you have them cheked for any possible clogs of blood and if they hurt you should consult a vein specialist for removal....ther r many ways to remove them now....check out vein centers or online at veinstop.com for complete and accurate info.