Can you avoid contracting the disease in Hole in the wall Fallout 4 quest?

You will contract the disease if:

• You get hit by a mole rat
• Your companion gets hit by a mole rat
• The protectron you can activate via terminal gets hit by a mole rat

Long story short, don't use a companion (or have them stay by the entrance), don't activate the protectron and Don't Get Hit. I was able to get through without catching the disease by abusing VATS and quick saves.

Note that the mole rats can be hit by grenades even while underground. If you get ambushed and have to reload, toss a Molotov Cocktail at the ground where the Mole Rat would appear and roast it early.

Also note that burrowed mole rats Will trigger mines. I tried to use mines on top of where they would spawn and blew myself up.

Here is some information about doing this on survival mode.

General tips:

• Just before you get to the ceiling turrets there is a bed you can use for saving. I was not able to save in any beds past this point.
• The secret part of the vault has some great loot. I brought all the good stuff back to the save room and stashed it in one of the dressers. When you are done with the quest, you can make a few quick trips back through the reactor room to get your loot.
• There is a drinking fountain in the hallway just past the save room if you get thirsty.
• Bring a lot of frag mines, frag grenades, and/or molotovs. It makes everything a lot easier.
• Bring a fast-shooting weapon with a reasonable clip that can 1-shot-kill a mole rat. I used a combat shotgun.
• I did this on survival at about level 30. I did not have a very stealthy character. Your mileage may vary.
• BE VERY SURE TO CHECK IF YOU HAVE MOLE RAT DISEASE BEFORE YOU SAVE

Walkthrough:

1. There wasn't anything in the first section for me. Just move on through until you get to a room with a terminal and a bed. This is the save room.
2. Just outside of the save room, there are two ceiling turrets and a protectron. Kill them and save.
3. Keep going until you get to a staircase down to a cafeteria. Run down the stairs and hide in the cafeteria. You don't need to sneak since the turret won't activate very quickly. Hang out until the coast is clear, pop out and kill it. Go back to the bed and save.
4. (optional) Activate and kill the protectron. If you throw mines down at the bottom of the stairs, it'll blow itself up. The protectron was a legendary for me and I wanted its loot. Go back and save.
5. Go down the stairs to enter a dirt area. There are some mole rats in here. Toss down a mine or two in the entrance doorway and in the passage into the room. There is an underground mole rat in the far left corner as you enter into the main chamber. It's kind of a cul-de-sac. Toss a grenade back there and race back out of the room to where protectron in step 4 is located. Try to not be stealthy so they come after you. In my game, the grenade killed the mole rat in the back and the others popped out of the dirt just before the entrance, blowing themselves up on the mines. I took out the last one or two. Save.
6. The dirt room ground floor should now be largely cleared of rats. Sneak back in and go up the stairs. Toss a mine on the stairs behind you in case anything pops out. Ditto on the next stairs that lead up to the door.
7. Toss a mine down in the doorway, and a few more down the hallway near where the room opens up. Find a good corner to nest in and toss a few grenades around to alert the rats here. Be careful to not detonate your own mines, though. Most of the rats will hit your mines. Some came from behind and some from the room in front of me. I only had to shoot one or two. Go back and save.
8. At this point, I ran around non-stealthy in the areas I'd cleared to make sure I'd killed everything.
9. Sneak back to door that leads to the stairs down. Open it up and lay a few mines around the stairs and in the room. Toss a grenade into the downstairs room to alert the rats and run back behind your mines. Most of the rats will die on the mines. Save.
10. Head downstairs to the dirt area. There will probably still be a rat or two down here. Toss a grenade or three around to kill them or bring them to the surface so you can kill them. When everything is dead, save.
11. In the next area, there will be rat cages/cells with the doors open. Rats will come out of them. You can either throw grenades into the cages or drop some mines, rile up the rats, and let them blow themselves up. I think my game had three rats here. Save.
12. You'll enter an elevated atrium area. The path to the right is blocked by crates and there will be an expert terminal you can open to bypass the rats and get to Curie. I wanted the loot, so I went left, where there a lot of mole rats. These seemed more powerful than the others and took a few hits to bring down. Again, grenades into the cages or leaving a mine trail will work well. In my game, one of the rats was glowing and had a lot of hit points. I used a critical to take him out.
13. At this point, all the rats should be dead so you don't need to save. You can just go meet Curie and finish the quest.