Normally we try not to answer in threads we moderate, in the interest of transparency, but given I just spent the last 10 minutes double-checking the stats to explain why the removed answer was terribly inadequate, I'd hate to let that go to waste for OP, so just sticking this up here as a disclaimer, of sorts.

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Anyways the draft was honestly pretty easy to avoid during the Civil War. To start with, you might simply not get called up of course. A draft is, after all, a drawing from the larger list of eligible names, and if you don't get picked, you don't get drafted. This could simply be the answer of course, that all three of those men were quite fortunate, but whether that was the case, we can't know. So assuming that they were the unlucky one's who had their name come out of the hat, what might have kept them out of Union Blue? Well, although 776,829 men were called for in the four drafts held between 1863 and the end of the conflict, it only resulted in 46,347 men actually conscripted into the military. Although all men between 20 and 45 were in theory eligible, there were many ways - both legal and not - to avoid the obligation.

To start, the single biggest way to avoid service was... to just not show up. 161,244 men whose names were drawn simply never reported for examination, a full 20 percent. In border states, including Maryland, so perhaps applicable here, this was particularly high, since Southern sympathies ran higher, and 30 percent of the draft evaders came from West Virginia, Missouri, Kentucky, Maryland, and DC. Further North, evasion often correlated with the proximity to Canada, or being particularly rural, so easy to avoid the provost marshals if they attempted to find you. Democratic strongholds also increased the likelihood, and areas where the population was heavily Catholic, or recent immigrants too showed a propensity to avoid service.

Next biggest group who avoided service were those who showed up for the examination and were given a physical exemption. Poor health, or some form of disability would preclude service, for obvious reasons, and 159,403 men were sent home for that reason. Almost as many, 156,106, were provided with various other exemptions which were deemed appropriate to keep them from service. Being the sole provider for a family was often enough, and quite a few were sent home for alienage (non-citizens. You were technically eligible for the draft if you were intending to become one though). Some names would be struck off because they were dead, and a small number who had reached 45 in the interim too would be exempted (see addendum below for a longer list).

Now assuming that you didn't didn't jump ship, and that you were determined to have a valid reason to avoid service, you still had options available. If you were a little flush with cash, a $300 commutation fee could be paid, which was availed by 86,724 men. Often, the fees would be put into a 'Bonus Fund' to be paid out to volunteers, who might receive anywhere from $50 to $100 for signing up (they didn't get the entire $300 of the fee!), on top of the existing Federal bonus. This proved to be a pretty unpopular means of avoiding service however, stoking accusations of class discrimination and the 'rich man's war, but a poor man's fight', in some cases the price was lowered, but and as such was only really available during the first two drafts, the option being removed in early 1864 for all but a select number of cases. While 52,288 men had paid the fee in the draft called on July, 1863, only 460 were allowed to in the final draft begun in December, 1864.

Similar to the commutation fee was the hiring of a substitute. Although sometimes conflated with Commutation, it isn't quite the same thing, that simply being the paying of money, while substitution meant providing a volunteer - or "volunteer" - to take your place, someone who wasn't included in the draft but was willing to serve. They would be provided a bonus by the person providing them in their place, and more than a few men simply made a career out of this, taking the bonus and deserting first chance they got, only to again accept a bonus down the line at a different location and repeat. Contrary to popular believe though, the substitution bonus was NOT set $300 (possibly a misunderstanding of the 'Bonus Funds'), it was whatever the two of you (or you and the broker, a thriving business at the time) agreed to, and in fact, the logic behind the Commutation fee of $300 was to ensure that the substitution prices would remain below that, although they did of course rise once that was eliminated. Anyways, hardly the rule though that they would immediately jump ship, and in total, 73,607 men avoided the draft by providing someone else willing to take their place.

For those keeping score at home though, we're still short a pretty large number from the total, because there was one further group, those who simply weren't needed! 93,398 of the men drafted were immediately discharged because the quotas had been met for their state or district, enough men prior having not been able to get out of it to the joy of those who shows up just at the deadline. This was especially true with the final draft, as the war came to a close, with 2/3 of those discharged even before examination coming from the final draft, simply not needed.

In fact, states which had met their quotas of volunteers simply didn't need to hold a draft at all. The draft as a whole was a substitution for the inability to meet recruitment numbers. Districts had a number of men they were expected to provide in the given term, and if they did so, no draft was needed there, so the draft was applied quite unevenly across regions. Your relative in Maryland, for instance, avoided the July 1863 draft simply because it wasn't called there as the had met their quotas with volunteers. Of course, it would seem that most men with an inclination to serve from Maryland were all in uniform by that time next year, as Maryland would then alone account for 57.7 of the draft evaders in the July, 1864 draft, 38,150 men simply not showing up when called.

So while without their records, we can't know for certain what avenue your ancestors took to avoid service, there were a number of possibilities. Assuming they were called up at all, and assuming too that they just didn't go, they might have enjoyed an exemption, or had the ability to pay the a substitute or else the commutation fee. Or they might just have been the lucky ones drafted so close to the end of the war they were told not to bother. The entire purpose of the draft needs to be understood more as an additional encouragement to volunteer, and the paltry numbers of actual draftees - 46,347 - compared to the 1,331,931 men who volunteered in that time (including the 73,607 substitutes, presumably) speaks to that fact. To be drafted was somewhat shameful, being seen as unpatriotic, and the looming threat certainly motivated a few men on the fence, who would prefer to definitely be an honorable volunteer than possibly a disreputable draftee.

Sources

Levine, Peter. "Draft Evasion in the North during the Civil War, 1863-1865." The Journal of American History 67, no. 4 (1981): 816-34.

Murdock, Eugene C. Ohio’s bounty system in the Civil War. Ohio State University Press for the Ohio Historical Society, 1963.

There were actually quite a few ways one could avoid being drafted, and two of them were supported entirely by law; those were substitution and commutation. There were a few other qualifications as well, which I'll get to below.

Substitution is what it sounds like; if you were called up, you simply had to have a volunteer take your place, and any arrangements between you and the volunteer were more or less up to the two of you. Most of the time this meant that the volunteer would be paid by the one whose place he was taking, but not always. Volunteers were paid a bounty by the Federal government, but it's unclear to me if this would include voluntary substitutes.

Commutation was simply paying a fee to avoid conscription, and the official commutation fee for the Enrollment Act was $300. This allowed the draftee to avoid being drafted for a year, and was a way not only to allow men to avoid service, but it also helped to raise considerable amounts of money for the war.

The problem is that it was so clearly biased toward the wealthy; a poor man, or even a man of modest means, would not have been able to buy themselves out of service, and so the unequal weight of the war fell mostly on the shoulders of the poor. This was not really a change from previous decades in specific, but the sheer scale of the effort to conscript men for the Civil War was much higher than any war previously, and the commutation law was deeply unpopular.

Another way to avoid service was simply to be married; because of some of the ideas about well-ordered societies, married men were considered vitally important to the stability of a culture, and so men who were married were enlisted only if the stakes got much higher. Married men could still volunteer, but they were often bypassed when it came to conscription.

Positions with the government or army that allowed them to assist in the war effort without shouldering a rifle could also have been possible. Sinecure positions were usually obtained through social standing or personal connections, and post-masters, rail depot supervisors, port authority agents and the like were exempt from the draft because their jobs were considered more vital to the war than simple soldiering.

In short, there were a variety of ways to avoid direct service in the US Army. If you know your ancestors' job and marriage status it might provide some insight.

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I tried and failed to find a full-text version of the actual enrollment act, but I did find a transcription of a NY Times article on the act and its popular effects here