It should work.
Presuming the spacebar works in other applications (otherwise that the keyboard should be fixed), you may try to close Illy and press Ctrl+Alt+Shift/Cmd+Option+Shift during startup.
When I imported my CorelDraw files into AI there was already spaces in the text.
I am able to edit this imported text and have no problem adding spaces using the space abr.
Its when I add new text I can't add spaces usinbg the space bar.
I reviewed the help menu and did find something about setting up the page through the charactor menu but I couldn't locate this action in the charactor menu.
BTW: the space bar works on other programs.
Your preferences may be corrupted, and the the suggestion in the first post, to close Illy and press Ctrl+Alt+Shift/Cmd+Option+Shift during startup, is a way of resetting them to the defaults.
You can be helped but you have to follow Jacob's instructions in post 1 and are you still on a PC (the windows system?)
You should always tell us about your system.
I tried the "Ctrl+Alt+Shift/Cmd+Option+Shift during startup" and it didn't work so I re-installed the program and that seemed to do the trick.
It must have been a bad install.
Thanks for all your suggestions.
For my part you are welcome, InvGrp.