Ultimately, Germany had little choice but to either do "something", or lose the war. They needed the resources of the Soviet Union, they needed to remove the growing threat on that border while it was still weak, and there weren't a lot of other options. Whether it was "possible" or not is highly debatable, and even at the time with the information available it seemed like a long shot at best to many of the German officers, but it was still probably better than inevitable defeat.
The disastrous failure to assess the true military strength of the Soviet Union is hard to fathom, but then again, how would you collect all of that scattered information. The Soviet Union was heavily decentralized, which makes it difficult for an insider to control, but makes it even harder for an outsider to spy on and get reliable information on anything outside of the immediate vicinity. There are a LOT of vicinities, and a lot less spies. Ultimately, the Germans destroyed more of the Soviet army than they thought existed. It wasn't even close to being enough. The ability of Stalin to prevent panic and the dissolution of the state under the stresses of losing those early battles was enough to turn the "improbable" into the "impossible" for Germany.
The initial stages of the invasion were a huge tactical success and a significant strategic one for Germany, but were simply insufficient on a Grand Strategy level. The German army pulled off a miracle, but found that several more of such miracles were required. The Soviets replaced their losses and just kept fighting. Other nations' armies reinforced understrength divisions in the field; Germany waited until its divisions were reduced to combat ineffectiveness, then pulled them back as a unit to receive replacements and rest. Until they reached that pathetic state, the Germans typically pulled their skilled technical and support personnel off their normal assignments in those divisions and threw them onto the front lines in desperation, where they were chewed up almost as quickly as raw recruits would have been. You can train and replace riflemen in a matter of a couple of weeks or months, but training and replacing the technical support elements takes years. Germany never recovered. In essence, Germany's armed forces "ate" their own logistical tails. While it can work to provide a final push in a short war, it's a recipe for disaster in a long campaign.
Note that Republican Rome's famed Triarii were on the same order as a more modern army's support elements. They included the armorers, paymasters, surgeons, carpenters, architects, accountants, clerks, and various other essential support functionaries (often with decades of experience and expertise) that allowed the unit to carry on in a variety of conditions on extended campaigns. They were NOT the elite fighting elements as often pictured in wargames, but their sizable pensions which were forfeit if the Legion were to be defeated and disbanded made them close to fanatical as a last-ditch defense. This was what Germany lost in the winter of 1941-42, which effectively made most of those battered divisions "green" due to the sacrifice of those key personnel. At one point, an exasperated Hitler commented that "This is not the same army as conquered Poland and France", to which his general replied, "That army lies buried in shallow graves across the Soviet Union".