I have terrible pain in the back of my leg when I walk. Could this be something serious?

Maybe. Pain associated with swelling can be related to a deep vein thrombosis (dvt), ruptured baker's cyst, muscle strain, tendon rupture or trauma. Pain in the muscle with walking that is relieved immediately with rest may be associated with artery blockage or stenosis (narrowing). Shooting pain down the back of the leg can also be related to sciatica. See your doctor to help sort out these possibilities.
Yes. Doubtful at age 34 but it could be a vascular blockage either arterial or venous. Possibly a neurological problem.