The option for hosts to record is now available for everyone on Android and iOS.
When starting a Space, tap the “Record Space” switch to have it available for public playback for 30 days once the Space has ended.
Once the Space is over, it’s available for 30 days after the Space has ended
However, the Twitter web app (on both desktop and mobile) still can’t be used to create a Space or record them — you can only listen to a Space hosted by someone else
The only option for hosting (and/or recording) a Space on desktop is to run the Android application in an emulator or virtual machine, such as BlueStacks
XDA » News Brief » Twitter rolls out recording feature for Spaces on iOS and Android