I recently purchased some 400watt MH gear and would like to know if it’s safe to run the lamp out of an enclosure. I do know that MH lamps have the tendency to sometimes explode apon EOL.

Running them vertically reduces the likelihood.

If it is an exposed rated lamp, then I would say yes this is probably safe.

But assuming it is an enclosed rated lamp (as most are), then I would say maybe. If the lamp is used (or you don't know it's history), I would not at all run it in the open. Technically even a brand new lamp can explode with a severe enough manufacturing defect, but I haven't seen that.

Explosion is always a risk, but if you shield the lamp or shield yourself you should be relatively safe even with an old lamp.

I recently purchased some 400watt MH gear and would like to know if it’s safe to run the lamp out of an enclosure. I do know that MH lamps have the tendency to sometimes explode apon EOL.

All HID lamps (excluding HPS, as they run at below atmospheric pressure) carry some nonzero risk of explosion. Even mercury lamps. In practice, it is rather minor for a short test run. Be careful with heavily used ones with blackened or bulged arctubes.

Surprisingly, old style high powered MH lamps like 400W multi-vapor tend to run at rather moderate pressures and are less risky than some smaller MH in 20, 35 and 70 watts!

There are open rated lamps having an extra protection shroud around the arctube for the case it explodes, which can be run with a peace of mind openly. Rare in high wattages.

There was a generic 6500K 400W elliptical clear MH lamp at Peamit Store, that exploded despite burned base-up in a highbay fixture: https://www.lighting-gallery.net/gallery/displayimage.php?pos=-251405

This is the lamp, it’s definitely not a shielded arc tube. It’s also new-old stock found in my churches basement after they got rid of their high-bays in favor of LED’s.

If your lamp has operating position restrictions, it is especially paramount to run it in its designated operating position such as VBU for base up or HOR for horizontal. If it is  rated for universal burning, it should safely operate in any position, but I recommend even operating universal burn lamps base up for maximum longevity.

https://www.scribd.com/document/132183345/GE-1984-Lamp-Catalog  Page 45 + 46 only say "ENCLOSED FIXTURE ONLY" if the 400w lamp is burned other than VBU or VBD +-15°.  Since your lamp is new , I see no problem ! The assembly plant I work in was lit by unprotected 400w M-59 lamps for 20 years. The maintenence morons NEVER replaced a lamp until it went out ! In that time I only saw ONE non-passive EOL in over 100+ lamps !

 Unprotected MH lamps were used for years . Yes , it's possible but unlikely for a MH to go BOOM ! Of all of the non-passive EOL lamps I've seen in the wild , a vast majority of them never penetrated the outer bulb .  The likelyhood of non-passive EOL increases dramatically as EOL approaches .

1% of EOL to be non passive is normally treated as way too much for applications where this type of EOL may cause significant damage or harm.

But thereal key here is evaluating what is the hazard of violent EOL in your exact use case scenario. If you are lighting an interior which could be set easily on fire or when people would very likely be within the range the whole time the light is used, then it would be better to take it very seriously.
But when the only thing that gets reached by the hot shower is a pile of waste scrap with nothing flamable all around, you don't have to care that much.  The protection measures in open rated lamps are by far not that effective as a very low probability somone walking into an otherwise no risk area just when the lamp goes.

And there is another measure that is quite effective in how violent the explosion becomes: Not letting the lamp burn continuously for more than few days. The thing is, the arctube gets weaker gradually. But also the warmyp cycle exposes it to extra stress. So when the arctube state approaches the level where it becomes more likely to explode, it will do that most likely during the warmup (when the various thermal gradient stresses add up), when also the internal pressure is not that high yet, so the explosion energy is way lower than it would be if it exploded during steady run.
This recommendation was often added into the open rating condition wording, usually along the base up position.