Being a SAHM is tough and the only way to cope is have structure and treat it like a job.
I'm not surprised you feel the way you are without a car and no other support - however, it isn't fair on your children and they must be aware of how you are feeling.
Look at it from their perspective - they have limited ways to communicate with you, want to have fun and have no idea of how annoying they can be.
I think for their sake and yours make some changes now.
Join the groups - there will be one for both, if not the 2 year old might have to be a bit bored and vice versa. I would rather my boys whinge and scream at a group rather than within our small flat - no one notices the crying and we're out rather than cooped up inside.
I have to literally get out every day - for e.g today, ds1 has chicken pox so we are limited, but we spent an hour this morning in the communal gardens of our flat picking up conkers and sticks. Back in the flat for lunch and an hour of cbeebies. They are both asleep now but when they wake up we will walk to the shop and post a letter. I could not cope staying at home everyday it would make me very depressed.
There's a charity called HomeStart - google them and see if they can help you.
Could you afford a cleaner? I employed one when ds2 (now 6months) was born - we literally signed up for the minimum - 2 hours every 2 weeks (£20) and said goodbye to our weekly takeaways. The boys love her and she is brilliant doing all my big jobs like oven, windows. This helps and I look forward to seeing her once a fortnight.
Is your 4yo not in pre school? You're entitled to your 15 free hours - use them!
Creches at gym for a couple of hours, soft play....
Sorry baby is now crying so have to dash - someone else will be along soon to give some excellent advice - hang on in there - they will both be at school soon and this phase will seem like a distant memory..