I gave up my car when I moved to the city and it's really quite do-able.
Online shopping is a must - I wangle mine on free delivery and find I actually save money as well by buying more precisely. (And have less shopping stress.) Milk and other fresh things can be obtained from local shops in between big deliveries.
Most journeys can be catered for by using taxis or public transport which being a city is quite good. The saving, even using the odd taxi or van hire, is quite large.
One tactic I would suggest, though, is working out how much you're currently spending on the car and setting aside, say, half of that each week in a 'taxi fund'. The problem is that car bills are so large that, surprisingly, you tend not to notice them (put on CCs etc) as much as handing over £10 cash for a taxi. A healthy 'taxi fund' which you can dip into without conscience for taxis, car hires, man with a van etc sorts that one out.
I'd also be very clear about lifts from family or friends. There will, inevitably, be the odd journeys which need a car these days and if you're offered or ask for a lift, I'd pay the friend or relative a very generous petrol allowance. I do that - and these days it's not refused. (10 years ago the offer might have been waved off but petrol is much more expensive now.) It's fair and it stops in their tracks any niggles that you might have 'given up the car just to use theirs'.
One of the great things to note is the removal of all the stress related to cars, parking, damage etc. You don't have to scout around for parking spaces and you can look with equanimity at parking issues outside in the street.
Worth a try I would have thought?