I'll try to keep this short. Basically, my husband's mother has never babysat DD1. When she has asked my husband, I asked him to politely decline. For one thing she is a smoker and not really all that stable and I have never really felt comfortable around her. I just feel that if I'm not comfortable around you then I'm definitely not going to leave you with my most precious gifts (i.e. my children). Now that DD2 has arrived, she seems to be pressing even harder to have the girls spend the night. I am still not comfortable with that, but now my husband is taking his mother's side. I have continued to say no, but now I'm wondering if I'm being irrational?
Re: I need to know if I'm being irrational...
If you are not comfortable with her and you are not comfortable with your children being around smokers, then no you are not being irrational.
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I don't think you are being irrational. I would never leave my children with a person that I was not comfortable around. In fact at this point in our lives, I would not leave my children with my own mother. Yet, I have no problem leaving DS with MIL+SFIL. (She watches him every single Wednesday and has had him overnight a couple of times.) This is based on the very different relationships I have with each of them.
I think that you do need to pinpoint why you are uncomfortable around her. I can see where your husband might feel slighted because his mother raised him, so he may not be able to see the bad in the same way you do.
I would also look at if you are uncomfortable due to a feeling or if there is a concrete reason. What do you mean by "not really all that stable"? I believe that if you can become clearer with yourself as to why you are uncomfortable that you will be able to better convey those thoughts to your husband.
(I am taking the smoking out of the equation for the time being because your husband seems to not see that as a valid reason. I believe that it is very valid reason to say no. I don't even care if she doesn't smoke while the kids are there. I strongly dislike being in any smoker's home and always feel unclean when I leave. I would not want my kids' to be any smoker's home for any extended period of time.)