After the treaty of Versailles, the German economy was thrown into shambles and there was instability throughout the country. So how were the Nazis able to create not only the army, but all the infrastructure needed with it? How were they financing all the research as well to have such top notch tech at the time and who was doing all the research?
The military of the Weimar Republic, the Reichswehr, provided much of the backbone for the German Wehrmacht of the Second World War. Although Versailles restricted its size, the planners of the Reichswehr created a highly efficient small army. The instability of the Republic in the 1920s meant that many Weimar leaders actually turned a blind eye to this contravention of the spirit of the Versailles Treaty as having a strong professional army could disband any insurrectionists. The danger of course was that as an established institution within the state, the Reichswehr could through its weight behind anti-democratic forces.
The Reichswehr's plan for expansion and rearmament centered around ensuring that the 100000 man army was fully trained and that Germany exploited any loopholes or lax enforcement of the Versailles Treaty. Given its small size, the interwar Reichswehr could afford to be highly selective in its recruits. Although the Versailles Treaty limited the number of officers, the Truppenamt (a shadow general staff) ensured that the ranks had a massive number of NCOs, which the Treaty did not prohibit. It operated under the aegis of Hans von Seeckt for most of the Weimar period. Von Seeckt instituted a training regime which incorporated many of the lessons of the last war. For example, the Reichswehr built many of its infantry formations around portable machine gun troops, the same type that found success in the late war Stoßtruppen. Officers and NCOs in the Reichswehr often thought and trained for the duties and responsibilities one or two grades above them, so that when expansion came, the German army would not lack for trained officers.
The Reichswehr also used a variety of means to evade the strictures of the Versailles Treaty. They fostered and maintained a Security Police of about 70000 men, often recruited from the paramilitary Freikorps. In order to develop experience with equipment forbidden by Versailles (namely aircraft and tanks), Reichswehr officers forged a clandestine relationship with the USSR. The state airline Lufthansa was also something of a shadow organization to get officers experienced with flying and cultivate air-mindedness. Finally, German military advisers were active in China aiding the Nationalist government in the 1920s to gain direct experience with military operations and put theory into practice.
To sum up, the Weimar era Reichswehr thus developed a solid foundation for a quick expansion. They kept this state of readiness throughout the entirety of the Republic. Hitler however expanded the Germany army far beyond anything von Seeckt or the Reichswehr planned. There is a peculiar irony in this. The Reichswehr at its inception had two competing answers to how it would respond to Versailles. Competing with von Seeckt's vision of a professional shadow army was a vast national Volksarmee of militias and paramilitaries. Eventually the professionals won out, but Hitler's expansion rekindled ideas of a national army based on conscription. The number of men under arms was far in excess for what von Seeckt and the Truppenamt planned for. Although the preparation of the Reichswehr paid dividends, the swift expansion often meant the Wehrmacht still suffered from a shortage of officers and its training regime suffered accordingly. Upper level strategy and logistics tended to be downplayed in von Seeckt's formulas since he was so focused upon expansion. The interference of Hitler and the politicization of the army only compounded such poverty of thinking on logistics and strategy. Moreover, the new military was quite short of equipment. Interim or training models like the Panzer II and I found themselves operating in frontline formations. The acquisition of Czech war industries proved a godsend to the equipment starved Wehrmacht, but the Czech equipment was approaching obsolesce in 1938 but the Germans kept things like Skoda's Panzer 38 on the frontlines well into 1941.
Edit: deleted double post and clarified language
Sources
Citino, Robert Michael. The German Way of War: From the Thirty Years' War to the Third Reich. Lawrence, Kan: University Press of Kansas, 2005.
Corum, James S. The Roots of Blitzkrieg: Hans Von Seeckt and German Military Reform. Lawrence, Kan: University Press of Kansas, 1992.
Murray, Williamson, and Allan Reed Millett. A War to Be Won: Fighting the Second World War. Cambridge, Mass: Belknap Press of Harvard University Press, 2000.
Well, works on public infrastructure and industrial efforts to modernize the army were exactly methods the Nazis used to revitalize the German economy. Prior to WWII, the German economy was -relatively- thriving. Through the "Volk" and "Volksgemeinshaft" rethoric they stressed autarky and nationalized important industries. Some historians claim that this method was only sustainable when it's efforts were funneled into a massive war effort. However, I'm by no means an expert on this period of time and you will have to wait for someone more knowledgable to complement or nuance.