I think you have some better options than having your cat euthanized by the end of the week.
You mentioned that she always wants to go outside. Is there a reason you must keep her indoors all the time? You might consider allowing her outdoors! Cats love being outside....there is far more in the great outdoors to keep them entertained, and that seems to be the main problem for you (space, and the fact that you can't give her the attention she deserves). To an outsider, that seems to solve your problem! At the very least, it would buy you some time to find a good home or get her on the waiting list at the Humane Society (which is a no-kill shelter). It would be very tragic to euthanize a perfectly healthy, loving pet because of lack of space & attention.