The problem is that I went to court 23 to get the x-01 power armor while I was level 25 and I killed the robots but left the armor because I wasn't high enough level. Now the doors to the robots room are closed and I can't click the buttons to open the door to the x01 armor. Please help.

Wasnt it court 35? And your gonna have to wait untill they respawn again

Yes, you're right. It is 35. How long until they respawn

Reload an earlier save from before you got anywhere near Court 35 and come back at level 28.

It'll be, and stay T-60 armor otherwise.

But I'm already level 41. Won't I lose all of that if I go back?

Yes.