The kart career [or I should say season] is really short. I think it is a total of 8 events each with 2 races 1 sprint and 1 rolling start with the grid based on the finish of the first race. If you set the race length slider to the lowest level then the first race is 2 laps and the next one is 3 laps. Some of the tracks take less than 30 seconds for a lap so it doesn't take long to complete every race in the kart season.
At the end of the season then you are faced with a choice as to which contract offer you want to accept generally in karts your choices are to keep racing the karts for another season or move up to the super karts.
The super karts are much more enjoyable than the karts as you run the super karts on normal race circuits and they are pretty fast >160mph and they stick too, very quick little things.
As for how long the career lasts, as long as you want it to you can run the same class multiple seasons or move to a different class. At the end of each season you will have options and of course you can have more than one career at the same time.
I started a new kart career yesterday for the Zero to Hero thing, drove a little yesterday and a little today and have already completed karts and super karts and the first event in Formula rookie.
I still have my other career which I started in Touring cars and am now in GT3 on that one