Have you read the expression here on Mamapedia, "One and done"? It's used by parents of only kids to express that they're finished.
I think you may have a case of "one and done" grandparents here. To be frank, after the first grandchild outgrew the sweet infant stage and another came along, they may (unconsciously, probably) have felt they were not up for two toddlers, which is where your family was inevitably heading. A nice attitude? No, but it's theirs to own and not yours to change. I agree with the person who posted that perhaps they came up with their furniture list as a way to telegraph to you that "You need to find another babysitter now." Again, nice way to tell you? No, but they possibly didn't even realize that was what they were really doing.
So, to the advice: Most important -- Do not vent to your mom when you are frustrated. You wrote that you called her while the kids were frustrating you, and vented to her, and instead of being the supportive sounding board you wanted, mom said what she said. So next time the kids are frustrating you, and you start to reach for the phone, call a friend, not your mom; accept that she is not going to sympathize and just offer support. If you call her again to vent, what's going to change about her response? Nothing. So I'd keep any discussions about the kids, with her, limited to fun things you did with them today, things they said, developmental milestones, etc. Because you know your relationship with her is not one that you can vent without YOU becoming more upset, not less.
I think too that others are right in saying she really meant to say "We'll be backup sitters if your regular one can't come." Nothing wrong with that. She at least is being clear (if clumsy) about letting you know where they stand -- willing backup sitters but not primary ones. Better to know that than to be calling them repeatedly and later hearing them say, "I wish she'd get that paid sitter back more."
You asked "How do you have a relationship with people like that?" Well, you do have a relationship. You're their daughter. Why not enjoy them as their adult child? When that sitter is over -- do things with your parents, without the kids. If they carp about "Where are the kids?" tell them, "This is our special mom (and dad) and daughter time." And go do something, and steer all conversation away from "Why isn't Suzy in preschool" etc. Surely you have other things you can talk with them about other than the kids?
They are not obliged to be your support system for your family's needs. Sorry, but there it is. Parents have to deal with things; grandparents do not.
My parents are both dead and my husband's parents are 6,000 miles away and both in very poor health. No other relatives within 5 hours and all the close friends work and care for either kids or aging parents. So the fact you even have parents around to speak to is good. But let their support be for you, their daughter, and accept that they are not obliged to be a support for your kids as well unless they really want it.