Plot hole or science faux pas in premiere episode of X Files season 10?

I've been reading up on Faraday cages in an attempt to answer your question. After rewatching the scene in the episode, I notice that first of all, the exact purpose of the Faraday cage isn't explicitly stated although the obvious conclusion would be to avoid RF detection and protect against electromagnetic pulse.

For reference here are a few images of the Faraday Cage from the episode, including the door that allows them to enter.

So is this a Faraday Cage? Can a Faraday Cage have a door?

I found this article to be a great source for the basics of Faraday cages without a lot of extra science mumbo jumbo.

Can A Faraday Cage Have Holes?

Yes, as long as the holes are small with respect to the wavelength of the incident electromagnetic wave. For example, a 1 GHz wave has a wavelength of 0.3 meters in free space. As long as the holes are significantly smaller than that dimension (i.e., a few millimeters), they won’t let in much of the incident wave. This is why fine conductive mesh can be used when constructing a Faraday cage. In practice, the cage’s lid or door usually causes the most leakage. Taping the seam with conductive tape helps to reduce this leakage.

So there can be holes but adding a door tends to cause leakage...

Here is another source indicating an almost identical response and example from different site:

Your Faraday cage can have small holes in it, provided they are not too large with respect to the wavelength of the incoming electromagnetic wave. This is why you can also use fine aluminum mesh to build a larger Faraday cage. For example, a 1 GHz wave has a wavelength of 0.3 meters in space.

Generally with these kinds of mesh cages, the cage door is typically the part that causes the most leakage, but this can be fixed by taping the seams with conductive tape.

So in answer to your question, yes this does look like a legitimate Faraday Cage and meets all the characteristics of one, however, you are correct that each time the door is opened they are likely creating leakage leading to them possibly being discovered and they would have been better off to creating a sealed entry and tunnel.

EDIT: I decided to go further with this in regards to the size of the holes in regards to allowing RF frequencies. Here is a chart of the US Frequency Allocations indicating the smallest to largest frequency channels.

Based on that chart I calculated the wavelength size in meters for the smallest (3 kHz) and largest (300 GHz) frequencies using an online calculator.

3 kilohertz = 9993081.933333 wavelength in centimetres
300 gigahertz = 0.09993081933333 wavelength in centimetres

It seems clear that no 3 kHz signals are getting in or out of the cage, however, 300 GHz have an incredibly small wavelength size and its conceivable that these wavelengths could penetrate a mesh cage as we see in the episode. So if the ARV was giving off signals of these frequency then its possible it could have been detected through such a cage.

The Faraday cage would have been more important to protect against inadvertent transmissions that could be detected. Thus, opening the door could be allowable if one took the precaution of first powering down anything that might cause that. Second, the cage doesn't have to be hermetically sealed... I'm not an RF engineer, but openings are allowable if they're some fraction of the size of the frequency that they're intended to block. And for exotic technology as depicted in the show, it's hard to say just what that frequency might be.

That said, I do not think that the X-Files is a show that the producers make much effort to have a consulting engineer/scientist vet the props/script.

Yes, and maybe not.

For it's intended purpose of keeping the ARV completely shielded from external communications or communicating externally, when you can't control RF emissions, having such a large door without a double-door setup does defeat the purpose.

But there is considerations for length of time the door is open, size compared to the rest of the cage, or the wavelength they know the ARV operates on. A partial Faraday cage will still attenuate a signal considerably. It won't be 100% effective, some signals will still get out, but the effective power and distance of these signals will be much much weaker. Even a kid can find that out:

Decreasing the Faraday cage enclosure configurations or shielding resulted in a progressively cleaner sine wave of the broadcasted tone, and a higher RMS voltage. I observed that signal blocking with a fully enclosed cage and one with only the top removed is about the same (90-100% shielding). I also observed that a three-sided enclosure can block approximately 85% of RF radiation, and that removing the bottom of a cage significantly increases signal strength, as compared to removing the top.

And they are in a metal bunker/hangar/warehouse, which itself is a partial Faraday cage, and helps attenuate any signal in or out. While not an ideal cage, this does set up a double cage of sorts.

So in-universe, any engineer that designed this cage likely took a look at all circumstances and rated the cage at a certain percentage of efficiency that was deemed acceptable.

The real reason is likely out-of-universe, to let the camera crew walk into the cage behind the actors without any complications.

There's a couple of additional uses for Faraday cages that haven't been considered.

1. The Faraday cage was used to block Remote Viewing, or at least limit it. The Far Sight institute talk of such a device in one of their documentaries, and there are ways to get around this for skilled Remote-Viewers. Maybe this is how they found it.
2. The Faraday cage was used to block Psychic noise allowing any potential pilot to practice controlling the craft in silence. I personally think this is more likely the reason, being a trans-dimensional craft it would need a highly sensitive (Psychic) pilot to drive it.