First off, our expectations of what the United States can accomplish in peacetime is coloured by a wholly modern phenomenon: the gigantic US war machine--well-trained, well-equipped and ready for conflict at short notice. This is actually a post-Korean War creation. Prior to this, the standard US military policy was to raise an army as needed, then almost entirely demobilize it when the crisis had passed, leaving the skeleton force with minimal funding. This was because, despite breaking from Britain, the US was built on a great deal of British philosophy and political theory. In particular, they inherited the British distrust of standing armies, like the British seeing them as too easily turned into an instrument of tyranny.

As such, when the American Civil War broke out, the US Army as a whole was very small: about 16,000 men north and south. 20% of the officer corps of this army immediately resigned on the outbreak of the war and pledged to the Confederacy. So, Union troops were small, under-supplied, and scattered all over the states and territories, and the desertion of officers reduced their ability to react and organize. This gives southern forces time to get their act together and mobilize troops of their own. The south had little trouble finding untrained bodies at the start, operating on a volunteer basis until 1862, because the war was brand new and many were eager to fight for the cause (in marked contrast to the situation north and south later in the war, with large desertion rates and draft riots). Unlike a navy, it doesn't take a long time to put together an army at this stage in history if you're willing to accept that you're fielding a green force incapable of complex maneuvers and hard marching, one vulnerable to breaking under even slight pressure: these are exactly the sort of forces that met at Bull Run.

As for "being in fighting shape", this is relative. The Confederacy was able to form a military, and it was ready for battle at Bull Run (with the caveats given above). Some of the goods they used were looted from federal storehouses in southern territory, providing a quick source of supplies. But a general theme for the south was scarcity: of firearms, of artillery, of food, even of uniforms and shoes (and of course, at the end, men). They improvised, they built up industry as fast as they could, they incorporated captured Union supplies, they smuggled in materials from the outside world past the Union blockade. Their armies would generally be poorly equipped and supplied compared to their northern counterparts, but as long as you have enough to do the basic job, a great deal can be accomplished; material superiority alone does not always win battles. Good leadership--something the Confederacy was often blessed with--could mitigate material inadequacies, by striking where the enemy didn't expect and thus breaking the morale of their forces instead of defeating them through material-based attrition combat.

Hey something I can actually contribute to!

You would get a lot out of Adam Goodheart’s 1861. That is where I am drawing most of this from.

To say that the CSA started from scratch is not exactly accurate. During the 1860 election, the country saw the development of a bunch of paramilitary groups. The first and perhaps most successful one were the “Wide Awakes”. The Wide Awakes were essentially the paramilitary wing of the Republican Party. They wore uniforms, were drilled, and held massive parades. They protected Republican politicians, rallies, and occasionally got into fights with Democrats, pro-slavery groups, and other anti-Republican partisans. In the South, similar groups quickly emerged, most notably the “Minute Men”. The Minute Men were drilled, armed, and prepared for an impending Yankee invasion. Obviously after what happened with John Brown's attempt to start a slave rebellion and the growing anti-Democrat, anti-slave power paramilitaries in the North, this actually seemed somewhat plausible to many Southerners.

There were a series of fires in Texas in 1860 that many believed was done by Wide Awakes or similar groups. Over a 100 black people and even some white people believed to have abolitionist sympathies were killed by mobs. In reality it was a heat wave. It was highly unusual for white people to be killed in sectional strife (north-south disputes). Prior to the war, obviously.

You also have to remember that state militias were a big thing at that time. They were on heightened alert after Brown’s actions. So the states of the CSA already had a large military pool of armed, trained, and dedicated Confederates to draw from when hostilities began. Of course, there were a lot of organizational issues, however.

I should also mention the role of Secretary of War John Floyd. Floyd is a rather obscure figure now, but he was a somewhat important figure in the lead up to secession. Floyd was President Buchanan’s Secretary of War, and a Southerner with secessionist sympathies. If Grant is to be believed Floyd in his capacity as Secretary of War deliberately sent weapons from Northern arsenals to the South in the lead up of secession. We're talking over 100,000 muskets. There is some debate about whether he did this because of the John Brown raid, or because he was supporting the imminent secession. However, he did order cannons to be moved from fortifications in the North to incomplete Southern military installations during the Fort Sumter crisis. So that would suggest treachery. The cannons at least was stopped from being moved when some concerned officers alerted the president.

At any rate, thanks to Floyd’s machinations, those groups I mentioned earlier were surprisingly well stocked. Floyd left his office during the Fort Sumter crisis and became a Confederate general. He died in the war.

I am getting some of this from the Personal Memoirs of Ulysses S. Grant.