No, we are not in a binary with Sirius. Our solar system's velocity relative to Sirius is many times the escape velocity of the Sirius system at our distance from it. The radial velocity of Sirius relative to us is 7.6 km/sec away from us, not to mention a proper motion of 1.2 arcseconds/year in declination and .546 arcseconds per year in right ascension (15 km/sec and 6.8 km/sec respectively). Since we're about 8.6 light years from the Sirius system, and since Sirius A and B have a combined mass of nearly 3 times the sun's mass, the escape velocity from Sirius at the distance of our sun is about 0.1 km/sec. In other words, our solar system is traveling too fast relative to Sirius in every direction you can measure for us to be in a binary with it.