These suits are used for traveling in the inhospitable vacuum of space. It is classified as light armor and can be repaired with radiation suits. It also has a high radiation resistance.
This armor is found in the REPCONN test site during the quest Come Fly With Me. It is found down in the REPCONN basement, overlooking the inaccessible irradiated rocket hangar, where one goes to give the fuel and thrust control modules to Chris Haversam. The space suit and helmet will be on a shelf to the left when entering the door. Unlike the suit, the helmet does not provide additional radiation resistance.
The only other space suits are worn by Bright followers, who will be found in the irradiated hangar. By killing them, extra space suits can be procured.
Like the advanced radiation suit, the space suit's Radiation Resistance effect is listed as "adv. radiation suit" in the status section of the Pip-Boy. The space suit, however, has higher DT and item HP.
The space suit helmet can be repaired with a head wrap, even though it is made purely out of glass and metal.
The helmet has several advantages: it does not decay, weighs nothing, and adds a decent amount of Damage Threshold. Also, one can still wear items such as the rebreather or other eye-wear while it is equipped.
The space suit (and helmet) can sometimes glitch, not allowing the player character to wear them due to the text blacking out. Dropping the equipment and picking it up again seems to fix this. [verified]
Equipping the space suit will permanently remove the Pip-Boy glove. This can be solved by using the console command player.equipitem 00025b83. Since this is just an appearance malfunction, you do not need it at all. Equipping other armors will display that armor's corresponding left glove instead. [verified]
An unlimited supply of this suit and helmet can be obtained by silently killing one of the Bright followers as they make their way to the basement. Once dead, search the dead follower's body and take the suit and helmet. After completing the quest, go back to the body and the space suit and helmet will appear again. Sometimes, the suits and helmets cannot be worn, but they can repair other suits or be worn by companions. [verified]
Attacking with a two-handed weapon (i.e. super sledge) in first-person view will cause part of the oxygen tank on the back of the suit to clip into the HUD. This is easily viewed with the camera pointing at a ~50 degree angle to the ground. [verified]
Rotating the camera around the player character in third person mode while wearing the space helmet will cause the face/hair of the player character to glitch, changing the facial features entirely as well as making them appear bald. [verified]