Would it be possible for a race of crustacean humanoids to live as nomads in a desert environment? I want them to have crab-like faces and crab-like skin with humanoid bodies. Would they survive in that environment? What would they need to survive both biologically/anatomically and clothing/equipment wise.
True crustaceans would have a hard time living in a desert environment since they have gills that need to stay moist at all times. For example look at the distribution of woodlice, which are our own version of terrestrial crustaceans. They live mainly in damp environments like under logs although there is apparently a desert species.
But you could handwave this without that much issue. Your nomads would need to trek at night and from burrow to burrow, foraging along the way or finding food underground. You'd need quite a lot of fossil water to keep the burrows or caves damp, maybe a Karst-like landscape.
As for equipment, something like a damp towel wrapped around the air-vents could give the nomads some more capacity to deal with heat and drought, I'd imagine them being in use by the hunting class or during emergencies, something high-status maybe.
It sounds like a unique and challenging environment, good luck!
If you mean a hot desert, a crustacean as large as a human would cook during the day as it would have no way of regulating its body temperature being cold blooded made worse by being encased in a giant exoskeleton. In order to survive it would need water perhaps a large oasis which it could climb into to avoid the temperature excesses. It would be able to emerge from the water a few hours before and after dawn and dusk and perhaps at other times for a few minutes.
Movement at night would also be problematic due to the low temperature, but slow movement might still be possible.
In order to survive on the land during the heat of the day the creature would need cooling, perhaps by painting its shell white and perhaps also in the form of a large sunshade. Large volumes of water would also be beneficial which it could spray on itself to provide evaporative cooling.
If you are willing to walk a bit up the family tree from crustaceans to their phylum arthropods you can find a number of insect and arachnid cousins who have adapted to live in desert environments. I'm thinking scorpions specifically.
The biggest problem you will face is likely based on size rather than specific environments. Exoskeletal creatures are limited in how big they can be due to a number of reasons such as; limited oxygen absorption without lungs, underdeveloped circulatory systems, structural support for large creatures without bones, etc. The largest arthropod (exoskeletal) creature currently is the coconut crab which can get to a respectable 1 meter across, but only weigh about 4 kilos. You would have to overcome these structural problems first for your large humanoid crustaceans to exist at all let alone adapted to a desert environment.
As the others have said, the biggest problem about crustaceans in hot and arid environments is that their shells do not allow them to release heat. On the other hand their shell could also stop most of the heat from going in.
It is hard to say, since we do not have many dryland crustaceans and even other life forms still try to stay inactive during the day. But I think that a less heavily armored body could be plausible, if we take a scorpion as an example(it's an arachnid, but it is probably the closest thing to a crustacean that lives in deserts). A scorpion is able to deal with heat, but it is still armored. So reducing the thickness of the plates and giving them a bright colour and adding some cooling parts should do the work.
Here is what I envision: a round, nearly spherical body. This would minimize surface area to volume and so be the slowest shape to heat up. The same high volume to surface area will minimize water loss. Appendages will be very long and minimally perfused - that gives you the height and size you want without increasing volume much. Basically it will be a head with long spindly legs and arms. This converges on the body plan of the harvestmen or opiliones, a distant cousin of your crustaceans but with a pretty crustaceanish face.
These things live everywhere, including the desert. You could have your alien versions wear flowing Bedouin like robes. Even better, though, would be for them to go naked except for very large and light sombreros.