They don't need a spacesuit just oxygen

Warning - Video of a spider in a vacuum chamber (it walks around and lives to tell the tale).

https://youtu.be/tA9jcIwvge0?t=243

Judging by the video, they don't even need oxygen for a short space walk.

In response to temperature concerns

It really depends where in space they are. Right next to a star they would be burnt up regardless of a space-suit. Away from any stars they would be frozen solid eventually. Their spacecraft could perhaps beam infra-red at them.

However:

At the Earth's distance from the sun, a space thermometer with roughly
half its surface is absorbing sunlight would register 45 degrees
Fahrenheit.
https://www.space.com/14719-spacekids-temperature-outer-space.html

Also many spiders can survive temperatures below freezing.

"...About 9% of spiders in relatively mild areas remain active
throughout winter. These are nearly all Linyphiids who can still make
a web at temperatures as low as -1°C. ..."

https://www.amentsoc.org/insects/fact-files/overwintering.html

Bonus fact

When they go on space-walks, because they're not enclosed in a suit, they don't have to worry about a connecting cable - they can use spider silk to hold them in the place they need to be.

I'm going to go in a different direction here and say that instead of making a space suit, make a space pod with mechanical spider legs.

Have an enclosed pod that the spider can sit inside with legs folded and have mechanical arms/legs that support the pod and allow it to walk around. For simplicity, the pod could have 4 or 6 legs, but there could also be different pods for different jobs with a variety of legs or attachments.

This has the benefit of looking like a spider suit, but doesn't have any of the complexity of trying to allow them to use their own legs.

Hamster ball

A hamster-ball for spiders allows them to breathe through their body, and move around by rolling like a hamster in a ball. Of course a simple plastic ball wouldn't allow much work to be done, but a high-tech ball could have various mechanical appendages to perform any work a space spider needs to do.

Tarantulas have retractable claws, similar to that of a cat, so it won't be a far stretch to think a spider could use these claws to fit a suit on.

Tarantulas have a respiratory system that uses book lungs to process oxygen. Tarantulas process oxygen at extremely low rates.

The hairs, or urticating bristles, are a defensive mechanism, early warning system(think Spidey Senses) and even a survival mechanism(some tarantulas use these bristles to create an air bubble around them, giving them potentially hours under water without the need to resurface for air).

Tarantulas are generally measured in diagonal leg span(from the tip of the front leg to the tip of the opposing rear leg).

Tarantulas generally become less active in the cold. Often not eating or drinking for months, or possibly years at a time.

Looking for videos of "tarantula molting" might give you some ideas how they could get the suit on/off. Molting is the process of shedding the old skin.

With all the above information in mind you will need to consider;
Are the suits needed for short space walks or would the Tarantulas be wearing them semi-permanently? This will determine if the suit needs air, heat and durability to elements).

I would design a suit where the only opening would be on the under side of the Tarantuals carapace. Allowing the Tarantula to fit it/remove it in a similar process to how it would molt. The fabric should allow the bristles to 'poke' out. The suit should retain warmth and small amounts of oxygen.

Technically, there are two main challenges with space suits.

1) Keep them loose enough and yet tight enough. Human limbs have bulgy muscles and such, to too tight and it's awfully uncomfortable. Platemail or plastic tubes wouldn't really work. For something that has an exoskeleton, this isn't a big deal, you're just duplicating that, essentially making the exoskeleton thicker.

2) Joints. Again, this is a challenge, but not too hard. Unlike with humans, all joints in a spider are relatively simple. There are two types: ball-and-socket, and hinge. Hinge (all the knees) are the simplest possible joints in engineering terms, but ball-and-socket (where legs join to body) are a ginormous pain in the ass in every imaginable engineering aspect.

Not impossible, just so prone to failure. Rather than a ball and socket, probably best to make two hinge joins in a chain, at rightangles to each other. Almost as good as a regular hinge, with most of the advantage of a ball and socket.

[Edit: it has been pointed out to me that arthropods don't have ball-and-socket, anywhere. They all do the two-hinged-joints thing everywhere, already, as I should have remembered from seeing crabs-legs. This makes exoskeletons insanely easy!]

So what it should end up like, is basically just a second skin. Spiders and other arthropods are already accustomed to wearing second skins, and are used to getting out of them, as we see here:

https://i.imgur.com/9K95I6E.gifv

You need to step back for an easier "in". I said in my comment to @Joe Bloggs's answer:

Based on how "dull" modern humans' senses are in comparison to our ape
cousins, I'm gonna say no. We're not even all that far from
chimpanzees, and [by comparison] we have hardly any sense of touch, smell, taste, or
hearing. Sight is the only comparable sense.

Compared to other
mammals, we're practically cut off from the world.

It's hard to
imagine a technologically advanced spider being just as endowed with
senses as its more primitive ancestors and contemporaries.

To answer the question based on that evolutionary assumption, then, you need to answer some questions about what these spiders look like, and how they feel about nakedness. I'm not talking homburg hats and four pair of Docs, but do they have any form of clothing? What are the names of the garments?

Anyway, to practical matters:

• Their hairy legs can be practically bald; and wearing a suit would only be as irritating as wearing socks in bed is for me, i.e. a little bit, nothing like unbearable or overwhelming. Sure, let's say they might need to put on some form of undergarment for protection.


• Their elbow/knee ears could be fed audio communications, designed into the suit, or the undergarment.


• Multiple eyes could be used for different purposes, assuming the minor pairs haven't atrophied by evolution.


• Breathing wouldn't be an issue. Spiders and other arachnids breathe through a book lung. Human space suits are pressurized. (If you have gas, you're breathing that back in.) I would assume that spider space suits would also be pressurized. The book lung of the space spider could also have seen modest evolutionary improvements to increase the airflow.

Everyday clothing is going to be your clue for how to talk about the different components of space suits, and what limitations they might impose on the wearer.

There is also the problem of the spiders' home atmosphere composition. Earth's has changed over the aeons, so sure, pick something. Maybe "they need more oxygen" which is fine because "they're super-strong". Just make your excuses and run with them. Don't necessarily declare them to the reader, unless they need to know world/space mechanics.

A couple of objections raised by others to my comment elsewhere:

1. the human sense of smell may be physically as good as that of dogs, but just out of practice




2. humans can feel surface bumps which are just 13nm high