A meltdown at a reactor like Chernobyl or Fukushima is not concealable, period, as several other answers have eluded: radiation will get out, and radiation is both very carefully monitored for by countries all around the globe, and the signature of the radiation (in terms of what isotopes are released and in what ratios) very distinctly identifies the type of nuclear installation that suffered the incident.
But in true Mythbusters style, let's reframe the question: can we adjust the circumstances so that a cover-up would be possible?
The black ops department responsible for managing Area 51 might have limited resources now, but back when this reactor was built at the height of the Cold War it would have been a different story. An evil-genius-style subterranean installation with blast shielding to resist an external nuclear strike? Yes please. Fully separated air reprocessing system with full isotope filtering? Where do I sign? The American military was big into nuclear hardening and keeping the nasty airbourne byproducts of nuclear war out of their installations; it's not too much of a stretch to think that those same systems would keep nasty airbourne byproducts of nuclear meltdowns in.
Of course for that to be plausible we have to not do too much damage to the infrastructure of our installation with the meltdown, but as noted nuclear reactors do not explode on meltdown, they melt, then usually catch fire, then set everything near them on fire. If there is an explosion, it's usually the rupture of an overheated (hence overpressurised) cooling system, which while very destructive to anything 'delicate' (delicate here including things like inch-thick steel pipes) won't break through metres of concrete rebar. It looks like people are still arguing about exactly what happened physically in the Chernobyl meltdown and how much energy was released, but it's clear that it's in the order of a few tons TNT equivalent, in the range of thermobaric bombs or large mining disasters.
With a yield like that, and the highly toxic airbourne byproducts, killing 6,760 people in the facility is entirely plausible, indeed it would be more remarkable if anyone survived. By containing the radioactive debris underground you essentially guarantee that anyone who breathes that air, dies, probably within hours. As noted, covering up the deaths of several thousand military personnel is going to be challenging. You can't realistically hide the fact that they existed or that they are now dead: it's not plausible that none of them had families or that they were all in 'deep cover': most of them will be technicians, cleaners, guards, doing ordinary(ish) jobs not in the rooms where the alien tech is stored.
However, in a high-security installation it will be possible to control the flow of information from the event and spin it basically however you like. Apparently it turns out that Corporal Surname-That-Sounds-Nasty-and-Foreign was a mole who launched an inside job to steal Perfectly Legal MacGuffin that the US was just being completely prudent and reasonable in guarding; in the process he released a Terrifyingly-Named Chemical Weapon - no, of course that wasn't produced by Patriotic Americans, it was supplied by That Foreign Power We Don't Like (let's slap some (more) sanctions on them in retaliation!), but it poisoned thousands of brave servicemen-and-women in the process. Colonel Strong-American-Surname-Whose-Family-Has-Impeccable-Media-Credentials bravely sealed the facility and fought the insurgents to ensure the safety of the MacGuffin and contain the chemical weapon, at the cost of his own life; his grieving family get a posthumous Medal of Honor or DSC, meanwhile a Navy Seal team shoot up a likely-looking villa in a far-flung country to get the Evil Mastermind behind the attack. Regrettably all the victims have to be cremated on-site, but families are flown in at the government's expense to attend the funerals - most of which can even be open-casket.
That will probably hold for a while, perhaps months or years. The biggest danger to exposure is probably radioactive contamination in the groundwater.