I'm amazed by all the posts saying that YABU and you should just suck it up no matter how ill you are. I completely disagree!
I think you and your husband are a team and have joint responsibility for children. Even if you've decided that he will work to support the family and you will look after the children (a job that's just as important IMO), you still share responsibility for finances and he still shares responsibility for the kids.
So if you are ill he should be helping you get support even if he can't support you himself by taking time off work. Obviously it depends how ill you are, and if you not too ill you can probably make do... But if you're so ill you can't move?! Frankly I would be concerned that he still expects you to look after the kids in that state, for your sake and theirs.
Do you have any close friends or family who live nearby and don't work? If so I think they should be the first people to ask if you're ill and need a bit of help. If not then the solution should be that your DH works from home or takes time off if he can, and if he can't (or refuses) they you pay a nanny or childminder.
What his father says about taking time off work to look after dependants being "career suicide", that's bullshit. Why is he just following what his father says and not discussing it sensibly with you? Obviously different employers have different attitudes to flexible working and time off, but as someone said you can legally take time of for dependants / family emergencies.
Do any of your DH's colleagues work from home or take time off to look after dependants? If so I think he could do the same. And if not he could at least discuss it with his employer - the worst that can happen is say no. They can't fire him or make him redundant for asking!