How can I repair a carpeted floor partly worn down by an office chair?
Possible, but unlikely.

It's not unusual for chairs used like this to create a ring of matted carpet where you usually sit, and the frequent pressure can damage the carpet's pad as well, giving the illusion of deep indentation. This is more pronounced with certain carpets.

The only way I can think of that this would seriously damage a subfloor is if you have moisture issues also (e.g. several drinks spilled, very moist crawlspace directly underneath, burst or leaky pipes nearby). Frequent or prolonged exposure to moisture will cause problems in any wooden subfloor.

Creaking can be a sign that the subfloor is damaged, but it's not unusual for some fasteners (nails, mostly) to move a little in some parts of older houses. If you take up the carpet and pad and see problems with the floor (water stains or rot are obvious indications) you might need to replace part of the subfloor - but again, this would be unusual and not directly caused by the chair.