The Pet (I can't be sure of the model, but my guess is the original 2001) is part of Kirk's antique collection.
Would it really still be working after 300 years?
It is there, in the background!
Year of feature (shown above)
"Would it really still be working after 300 years?" If they can transplant Spock's brain, they can certainly keep a Commodore running well into the 23rd century. ;)
Seeing as it's part of his antiques collection, I'm sure Kirk himself would keep it working.
May have been an inside joke as Shatner shilled for Commodore at one time.
Perhaps it was replicated?
Since it had blue text on black background, it has to be model 2001 being used in the movie. only the 2001 used this scheme. All other PET had green text instead.
The commodore could definitely still be working after 300 years. Hobbyists already replace capacitors/other components, and reproduction boards have been made of retrocomputers already.
The monitor would be the hard part - a CRT monitor would be quite the rarity centuries later.