That is, he doesn't just walk into a shuttle bay, commandeer a shuttle without anyone noticing (because these places are always totally devoid of personnel), get the shuttle bay doors open, fly out of the bay AND get it too far from the ship to be tractored in before the system so much as blips a warning.
When Roga Danar escapes from his cell (unbelievably but hey, at least it takes some effort), the bridge crew immediately shuts down all of the shuttle bays and transporters. Finally! Personally, I think it is really dumb that the transporters wouldn't be shut down anyway. I mean, where's anybody going to go? The Enterprise left the planet to go chasing after Danar. Do individuals habitually want to use the Enterprise's transporters to mock transport themselves to non-existent places? "Look, Mom, I'm transporting myself out into space. No, I'm not." Still, at least Danar has to outmaneuver this obstacle.
The bridge crew also set up forcefields on every deck. Again, well, yeah, people. Do ya think? It amazes me how often the bridge crew are stymied by wandering bad guys.* I'm not sure Danar should be able to get as far as he does in fact but by Star Trek standards, he's pretty smart so I'll allow him to outmaneuver all possible non-addressed-in-the-episode forcefields.
*Actually, it doesn't amaze me all that much. See below for my theory on why.
Anyway, Danar then has to outwit some form of tear gas; this is never fully explained. I think he is able to endure it because of all the amazing superman genetic manipulation he has undergone. He also has to outwit Data, which is somewhat unbelievable. (See my theory below for why it MIGHT work.) And okay, the ease with which Danar gets to Engineering just blows your mind. And the fact that no one runs to catch him. "He's headed to Engineering," Riker says. "On my way," Worf says and saunters briskly down the corridor. (Of course, if he ran, the camera would run out of room.)
Still--STILL--Danar has to work to get off the ship. It is, on the whole, a preferable episode to those episodes where the bridge crew only learns that someone is transporting (or borrowing shuttles) after the fact and oops, it's just too late to do anything about it. (One could argue that for all episodes where this happens, the viewer should assume that a long, Danar-like chase occured before Wesley or Worf or Data says, "He's transported to the surface, Captain!")
*My theory regarding Star Trek. My theory is that the Federation is run by a bunch of bureacratic types who honestly believe (or pretend to believe) that they aren't a military organization, like having extreme leftists run the war in Iraq. So they have all these bizarre rules and regulations that hamper the people who actually know what they are doing (actually, it's more like the British uppercrust bureaucrats trying to run India versus the lowerclass officers and engineers in India who actually knew what was going on; Kipling had scathing things to say about the British uppercrust types). So, they insist that shuttle bays not be guarded because that would imply, oh horror of horrors, that the Enterprise is a military ship (this theory also explains the reason Picard keeps telling everyone it isn't when it is). And they have to leave the transporters on because otherwise the Enterprise will be deemed a hostile, prison-like environment. And all that business about having one huge computer system so that problems on the holodeck always, always, always cause photon torpedoes to arm: that's the fault of the idiot bureaucrats as well. (Actually, Star Trek 3: Search for Spock more or less implies that the Federation is run by lousy bureaucrats).
My theory about Data is that his creator built into his memory processes a "slow" device. That is, the creator wrote a program where Data is forced to discard or randomly select information before he finishes computing all possible variables. It's really subtle; he can still rattle off information and numbers at the drop of a hat, but it keeps him from being too computer-like and forces him to learn: it's kind of like one of those gates in irrigation drains in Utah that flaps back and forth depending on the force of the water so the more variables there are, the more Data's response is "slowed" or checked (it's still very fast, it's just Vulcan fast rather than machine fast). Data's program also insists that he actually experience an "agency moment" where he has to think about his final choice rather than simply--computer-like--producing it. This checking-and-agency program makes it possible for Troi to beat Data at chess. And for Danar to "outcompute" him. And various other unbelievable scenarios to occur. It also explains the absolutely awesome ending of "The Most Toys" (great episode with ever talented Saul Rubinek).
All in all, enjoying Star Trek is much easier if one accepts the basic pointlessness of the Federation. Of course, Deep Space Nine got all gloomy about it. It is much more user friendly if one simply assumes that most of these people are totally clueless. We lucky viewers get to watch the few non-clueless Federation members. Well, non-clueless most of the time.