I think it can be both normal and hard. My life was similar to yours when my 2DC were small (they are 10 and 12 now). 2y age gap so when they were 0 and 2, 1 and 3, and 2 and 4, until the oldest was in school, these were tough years. I worked 3 days and DH worked long hours. He had some health and mental health issues and didn't / couldn't pull his weight. We were skint from childcare costs, no family help so rarely got a break. As with you, struggled to keep on top of housework and DIY. DD1 particularly was a liability and had to be watched every minute, never settled to anything due to short attention span - she frequently had me in tears - we found out later she has ASD.
In contrast to you, though, my job wasn't that stressful and I had been in it for years.
Some things that helped:
Doing fun things on my days off with the kids, usually meeting up with a friend with similar aged kids at least one day. So I did get some socialising even though it was chaotic with the kids. I also did some groups with the kids (eg drama, playgroup) which were again chaotic but gave the day structure.
When I was on annual leave, I'd sometimes keep the kids in childcare and have a lovely day to myself or blitz the house. It was only a few times a year but it did make a difference. Sometimes DH and I both had a day off together and went out for lunch etc.
If I was ill, I wasn't a martyr - I called in sick, stayed home alone and rested. This made a massive difference to how quickly I recovered - I found i couldn't power through like I would have done pre kids. I did get more ill than usual - maybe 5-10 days a year? - I guess because I was tired and run down.
Dh and I gave each other a break by taking the kids out / away at the weekend to allow the other one to catch up on house jobs / rest / whatever. There was a period when he didn't take both kids at once and I wish I'd pushed harder with that, but as I say, he was depressed and I worried about pushing too far.
I still did find that period very hard and tiring but I did appreciate my job (wouldn't have wanted to be ft with the kids) and also looked forward to my days off with the kids (again, v grateful I could be pt).
As pp say, you can't really go off sick long term, but if you think a couple of days off to sleep and rest will help you regain your energy, do that.