Records are kept in the National Archives in Washington.Many libraries have copies of the microfilm.Fold3 has the records on line for a membership fee.Other sites such as history-sites.com will send you copies for a fee.You do not see copies of the original muster rolls or other records.What you see is a copy of an individual's Compiled Military Service Record.These records were compiled in the late 19th and early 20th century from the rolls, so that it would be easier to confirm the service of an individual for pension purposes.
The two Hubbard Browns you refer to were, indeed, two separate persons.Private Hubbard Brown of Company F, 21st Virginia Infantry enlisted on 24 Feb 1863, in Campbell County.Enlisting this late in the war might be an indication that he was too young in 1861 or 1862.He was last listed on the muster roll of Oct. 31, 1864, with the following notation:"Absent - Missing since 19 Oct '64, supposed to be a prisoner."The following from a reenactor group's web-site is instructive: "General Early remained in the lower valley until the second week of August 1864, when Sheridan forced him to withdraw to Fisher's Hill. Early, however, received re-enforcements from Lee, and by August 17, had pushed Sheridan back across the Potomac. The Confederates were once again in the lower valley, where they remained for a month. Sheridan moved against Winchester on September 19, forcing Early to retreat as far as Fisher's Hill, where he made a stand. F Company had three wounded in the battle fought there on September 22. After Fisher's Hill, Early retired up the valley to Mt. Jackson. On October 19, he attacked Sheridan at Cedar Creek, but what was initially a Confederate victory turned into a defeat at the close of the day. Early was driven from the field, losing most of his artillery and many of his men as prisoners. F Company lost one killed, and three wounded, including Lieutenant Hudgins, who was captured. Cedar Creek virtually ended the 1864 Valley Campaign."
The other Hubbard Brown was a 1st Sergeant in Company H, 42nd Virginia.He enlisted May 22, 1861, at Spoon Creek, which I suppose was a community in Patrick or Henry County. Men from both counties were in the 42nd.He last appears on the muster roll for 31 Oct 1864, listed as present. No further records.
Many Confederate soldiers were buried in mass graves or on the battlefield where they fell.Comrades would mark graves if they could, but if they did not hold the field they could not.Efforts have been made to list the dead in certain cemeteries, or in some cases in entire states (Mississippi has such an index), but there is no one index of all known Confederate burials.