YANBU
It's very normal and natural to feel a bit wobbly when something really big like this happens, and in your case it was basically sprung on you by him so it's not something you planned, even though deep down you may have wanted it, and even though you are feeling much the better for it on many levels. Women are conditioned not to rock the boat, and when a relationship has gone on long enough to include four children the oldest of whom is 12, putting an end to it when he is dangling the alternative in front of you can feel like the mother and father of rocking the boat.
(This is why the OP posted, I suspect).
cheaperthebetter, you seem very aware of what he is doing here (chasing you) so I suspect he has played this game before (of creating distance and then coming back, even in an emotional sense) and he has every expectation that you will play along this time. It is no doubt coming as a huge surprise to him that you are standing your ground.
PLEASE STAY STRONG and do not be taken in by any promises of change, and especially, when he starts to get angry and starts to feel victimised here (as he will) please do not feel so afraid that you give in.
Pack any stuff of his that he left behind. Leave it outside. Tell him where it is. If you have a front garden to leave it, that's great. Make sure you have a friend or family member with you when he comes to pick it up (arrange a time). If there is a neutral place where you could have a friend or family member bring it to him at a prearraanged time that would be much better.
You need to talk to your family and friends about what has happened and ask for support. Tell them you are not looking for arguments about why he should be given another chance or suggestions that he is in a bad way now, nowhere to stay, whatever. Just ask them to back you up.
Set up a new email address just for him to use. Don't respond to any more calls or texts from him apart from the text you send telling him no means no, and the relationship is over, and giving him details about the new email address he is to use to contact you from now on.
I agree with those saying keep the DCs out of this. Sit them down and tell them what is happening but emphasise that this was your decision. Thank them for being honest with you about how they feel.
You could apologise to them for letting things go on for so long when they were miserable. I know his behaviour wasn't your fault, but a parent can teach a valuable lesson even in the least promising of situations, and this way you assert yourself as a protector of the children, which is important when dealing with a man like your stbx and it's important for the children to know they have a protective parent.
See a solicitor and CAB.
You need advice about contact with the DCs, about getting him off the lease if you rent, about who may live in the home (residential order) if you own your home, whether you can change the locks (a landlord might be willing to do this for you). Also about divorce proceedings if you are married.
Who owns your home? Financial advice is needed.