Why didnt the civil war have armor?

The American Civil War did see many experiments in body armor; Bashford Dean in Helmets and Armor in Modern Warfare 58-59 notes:

It is known that breastplates were worn more or less frequently during the American Civil War. In the museum in Richmond, there is preserved such a "suit" of armor, Fig. 15, which at the time of the siege was taken from a dead soldier in one of the trenches. He was shot in the side or back, for the breastplate, it appears, was not penetrated. This armor was of northern origin. Further inquiry shows that a factory for the making of such defenses was established at New Haven about 1862. The metal employed was a mild steel, .057 inch thick, and the "suit" weighed about seven and one half pounds. While no tests of this armor are available, we estimate from the thickness of its metal, assuming that it is a "mild" steel, that it would have stopped a 230-grain pistol ball traveling at the rate of 500 foot seconds.

What the American Civil War did not see was formal adoption and issuing of such armor to all troops; and with good reason. The South faced shortages of metal and material throughout the war, and couldn't afford it; the North probably could have afforded it, but it wasn't worth it. To quote Sarah Weicksel:

In theory, the vests were appealing. As one soldier wrote: "To be 'iron clad' when the bullets should fly as thick as hail! What more could a soldier ask?" (Walker, History of the Eighteenth Regiment Conn. Volunteers, 21). In actuality, however, the vests proved to be failed objects on multiple levels, ranging from ease of use to their effectiveness. Although advertisers claimed that the vests were "simple" and "light," soldiers found them extremely cumbersome due to their inflexibility and weight. Colonel Charles F. Johnson of New Jersey explained to his wife: "the only objection that I have to them is that they are so confounded heavy for this season of the year" (quoted in Pelka, ed., The Civil War Letters of Colonel Charles F. Johnson, 112). Many soldiers' letters and memoirs recounted the abandonment of bullet proof vests along the march, where they littered the side of the road along with other unwanted gear. [...] The vests did provide some degree of protection, judging from the bullet-shaped dents in surviving vests. But they were ineffective in close combat, and, as Johnson rightly pointed out, wearing a vest could have resulted in an even more deadly wound if a man was shot at close range, whether from immediate impact, or an infection festering around the bits of cloth and metal that the bullet pushed into his body.

These are more or less the same issues Dean noted that soldiers and military armorers faced in WWI (and why, he also says, armor was generally abandoned in the first place): against firearms, any kind of body armor that was thick enough to be of any protective value was, generally, too heavy and cumbersome to be born for long, and probably too expensive to equip troops with.

Wasn't close quarters fighting common?

Not really. While it is true that some fighting did undoubtedly come down to hand-to-hand and bayonet fencing, there were very few bayonet wounds during the Civil War, and fewer wounds from swords, the troops just rarely came in to that kind of contact. Robert L. O'Connell in Of Arms and Men: A History of War, Weapons, and Aggression wrote:

Despite frequent expressions of confidence by commanders on both sides, the bayonet played a minuscule role in the killing during the Civil War. Not only were most bayonet charges turned aside by rifle fire, but even when they did succeed, the defending troops almost invariably ran away before these weapons could be employed.