Really good question and you've had good answers. For me good saving starts with a really good budget, and if you've truly got your budgeting right then you should very, very rarely have the sort of unexpected expenses that wipe out your long-term savings - like a PP said the sort of things that can take you by surprise usually are foreseeable with some planning, e.g. unless you have a very new car that's under warranty you know it's going to need an MOT and servicing every year, you know how much mileage you do so how often/when you would expect it to need to get new tyres, replace the brakes etc and I also try to put away in a 'car fund' approx 10-20% of the car's (new) value every year into my 'car fund' which will pay for more major repairs and/or a new car when the current one dies. I may get lucky and get more use out of my car than that (in fact current car is an 09 plate and still going strong but I know it probably hasn't got that many more years left, so I am building up the fund for the replacement). So yes although at some point soon my car fund will go to 0 I wouldn't get stressed about it because I've been planning for this for years. Same with white goods, boiler, the roof of the house, the pets and their ailments/vets bills, pretty much all those big expenses that pop up from time to time I take a pessimistic view and try to be always building a fund (seperate from my actual/long term savings to be covering).
For things like Christmas and birthdays and holidays which come around every year but still take some by surprise, again I try and dispassionately but realistically decide in advance what my budget is then set aside 1/12th of that into a separate fund every month (online banks like Monzo are brilliant for this as you can make labelled 'pots' for each expenditure), then when the time comes around be strict with myself not to exceed the budget (or if I do that it has to come from somewhere else e.g. my personal spends or clothes, not long term savings).
Basically budget, budget, budget. As a rule of thumb, 50% of your expenditure should go on 'needs' i.e. rent/mortgage, bills, food, petrol/travel, insurances and I would include building a fund to cover annual bills and white goods replacement in this, 30% should go on wants i.e. hobbies and leisure, children's activities, holidays and birthdays, and 20% then to savings (you could include things like saving towards bigger house maintenance projects, car replacement in this as well as pensions and building investments). Now of course the formula doesn't work for everyone but if either your needs or wants funds vastly exceed the %s then it may be the time to think about either seeing if you can reduce expenditure (especially in the 'wants' category) or increase income in some way. Like I say that absolutely isn't possible for all, some people can cut absolutely everything to the bone and still be left living payday to payday and not able to save anything like 20% of their income so not saying this is the only way forward but I still think the principle of map an accurate and realistic estimate of your actual annual expenditure in all these areas will help you understand how much you can 'actually' save.